mmtheorems10 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 901-1000 - Page 10 of 108

Type	Label	Description
Statement
Theorem	3orim123d 901	Deduction joining 3 implications to form implication of disjunctions.
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ta))   &   |- (ph -> (et -> ze))   =>   |- (ph -> ((ps \/ th \/ et) -> (ch \/ ta \/ ze)))
Theorem	an6 902	Rearrangement of 6 conjuncts.
|- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> ((ph /\ th) /\ (ps /\ ta) /\ (ch /\ et)))
Theorem	mp3an1 903	An inference based on modus ponens.
|- ph   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ch) -> th)
Theorem	mp3an2 904	An inference based on modus ponens.
|- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mp3an3 905	An inference based on modus ponens.
|- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mp3an12 906	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ch -> th)
Theorem	mp3an13 907	An inference based on modus ponens.
|- ph   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ps -> th)
Theorem	mp3an23 908	An inference based on modus ponens.
|- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	mp3an1i 909	An inference based on modus ponens.
|- ps   &   |- (ph -> ((ps /\ ch /\ th) -> ta))   =>   |- (ph -> ((ch /\ th) -> ta))
Theorem	mp3anl1 910	An inference based on modus ponens.
|- ph   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ps /\ ch) /\ th) -> ta)
Theorem	mp3anl2 911	An inference based on modus ponens.
|- ps   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ch) /\ th) -> ta)
Theorem	mp3anl3 912	An inference based on modus ponens.
|- ch   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ps) /\ th) -> ta)
Theorem	mp3anr1 913	An inference based on modus ponens.
|- ps   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ch /\ th)) -> ta)
Theorem	mp3anr2 914	An inference based on modus ponens.
|- ch   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ th)) -> ta)
Theorem	mp3anr3 915	An inference based on modus ponens.
|- th   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch)) -> ta)
Theorem	mp3an 916	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th
Theorem	mpd3an3 917	An inference based on modus ponens.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpd3an23 918	An inference based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	biimp3a 919	Infer implication from a logical equivalence. Similar to biimpa 416.
|- ((ph /\ ps) -> (ch <-> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3anandis 920	Inference that undistributes a triple conjunction in the antecedent.
|- (((ph /\ ps) /\ (ph /\ ch) /\ (ph /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch /\ th)) -> ta)
Theorem	3anandirs 921	Inference that undistributes a triple conjunction in the antecedent.
|- (((ph /\ th) /\ (ps /\ th) /\ (ch /\ th)) -> ta)   =>   |- (((ph /\ ps /\ ch) /\ th) -> ta)
Theorem	ecase23d 922	Deduction for elimination by cases.
|- (ph -> -. ch)   &   |- (ph -> -. th)   &   |- (ph -> (ps \/ ch \/ th))   =>   |- (ph -> ps)
Theorem	3ecase 923	Inference for elimination by cases.
|- (-. ph -> th)   &   |- (-. ps -> th)   &   |- (-. ch -> th)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th
Other axiomatizations of classical propositional calculus
Theorem	meredith 924	Carew Meredith's sole axiom for propositional calculus. This amazing formula is thought to be the shortest possible single axiom for propositional calculus with inference rule ax-mp 7, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 4, ax-2 5, and ax-3 6. Then from it we derive the Lukasiewicz axioms luk-1 938, luk-2 939, and luk-3 940. Using these we finally re-derive our axioms as ax1 949, ax2 950, and ax3 951, thus proving the equivalence of all three systems. C. A. Meredith, "Single Axioms for the Systems (C,N), (C,O) and (A,N) of the Two-Valued Propositional Calculus," The Journal of Computing Systems vol. 3 (1953), pp. 155-164. Meredith claimed to be close to a proof that this axiom is the shortest possible, but the proof was apparently never completed. An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic somewhat late in life after attending talks by Lukasiewicz and produced many remarkable results such as this axiom. From his obituary: "He did logic whenever time and opportunity presented themselves, and he did it on whatever materials came to hand: in a pub, his favored pint of porter within reach, he would use the inside of cigarette packs to write proofs for logical colleagues."
|- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((ta -> ph) -> (th -> ph)))
Theorem	merlem1 925	Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.)
|- (((ch -> (-. ph -> ps)) -> ta) -> (ph -> ta))
Theorem	merlem2 926	Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ph -> ph) -> ch) -> (th -> ch))
Theorem	merlem3 927	Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ps -> ch) -> ph) -> (ch -> ph))
Theorem	merlem4 928	Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ta -> ((ta -> ph) -> (th -> ph)))
Theorem	merlem5 929	Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> ps) -> (-. -. ph -> ps))
Theorem	merlem6 930	Step 12 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ch -> (((ps -> ch) -> ph) -> (th -> ph)))
Theorem	merlem7 931	Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))
Theorem	merlem8 932	Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))
Theorem	merlem9 933	Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ph -> ps) -> (ch -> (th -> (ps -> ta)))) -> (et -> (ch -> (th -> (ps -> ta)))))
Theorem	merlem10 934	Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> (ph -> ps)) -> (th -> (ph -> ps)))
Theorem	merlem11 935	Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> (ph -> ps)) -> (ph -> ps))
Theorem	merlem12 936	Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((th -> (-. -. ch -> ch)) -> ph) -> ph)
Theorem	merlem13 937	Step 35 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> ps) -> (((th -> (-. -. ch -> ch)) -> -. -. ph) -> ps))
Theorem	luk-1 938	1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- ((ph -> ps) -> ((ps -> ch) -> (ph -> ch)))
Theorem	luk-2 939	2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- ((-. ph -> ph) -> ph)
Theorem	luk-3 940	3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- (ph -> (-. ph -> ps))
Theorem	luklem1 941	Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> ps)   &   |- (ps -> ch)   =>   |- (ph -> ch)
Theorem	luklem2 942	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> -. ps) -> (((ph -> ch) -> th) -> (ps -> th)))
Theorem	luklem3 943	Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> (((-. ph -> ps) -> ch) -> (th -> ch)))
Theorem	luklem4 944	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((((-. ph -> ph) -> ph) -> ps) -> ps)
Theorem	luklem5 945	Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> (ps -> ph))
Theorem	luklem6 946	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> (ph -> ps)) -> (ph -> ps))
Theorem	luklem7 947	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> (ps -> ch)) -> (ps -> (ph -> ch)))
Theorem	luklem8 948	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> ps) -> ((ch -> ph) -> (ch -> ps)))
Theorem	ax1 949	Standard propositional axiom derived from Lukasiewicz axioms.
|- (ph -> (ps -> ph))
Theorem	ax2 950	Standard propositional axiom derived from Lukasiewicz axioms.
|- ((ph -> (ps -> ch)) -> ((ph -> ps) -> (ph -> ch)))
Theorem	ax3 951	Standard propositional axiom derived from Lukasiewicz axioms.
|- ((-. ph -> -. ps) -> (ps -> ph))
Theorem	nicodraw 952	Axiom of Nicod from Introduction to Mathematical Philosophy B. Russell, p. 152. The axiom is recovered from this raw form by substituting (ph | ps) for -. (ph /\ ps), where | is the Sheffer stroke (NAND) connective, so that the Sheffer stroke becomes the sole connective. See nicodmpraw 953 for the inference rule. (Based on a proof by Jeff Hoffman, 19-Nov-2007.)
|- -. (-. (ph /\ -. (ch /\ ps)) /\ -. (-. (ta /\ -. (ta /\ ta)) /\ -. (-. (th /\ ch) /\ -. (-. (ph /\ th) /\ -. (ph /\ th)))))
Theorem	nicodmpraw 953	The inference rule for the axiom of Nicod, in raw form as explained in nicodraw 952.
|- ph   &   |- -. (ph /\ -. (ch /\ ps))   =>   |- ps
Predicate calculus axiomatization
The axioms of predicate calculus
Syntax	wal 954	Extend wff definition to include the universal quantifier ('for all'). A.xph is read "ph (phi) is true for all x." Typically, in its final application ph would be replaced with a wff containing a (free) occurrence of the variable x, for example x = y. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of x. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
wff A.xph
Syntax	cv 955	This syntax construction states that a variable x, which has been declared to be a set variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder {y | y e. x} is a class by cab 1463. Since (when y is distinct from x) we have x = {y | y e. x} by cvjust 1471, we can argue that that the syntax "class x" can be viewed as an abbreviation for "class {y