mmtheorems9 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 801-900 - Page 9 of 108

Type	Label	Description
Statement
Theorem	3ad2ant2 801	Deduction adding conjuncts to an antecedent.
|- (ph -> ch)   =>   |- ((ps /\ ph /\ th) -> ch)
Theorem	3ad2ant3 802	Deduction adding conjuncts to an antecedent.
|- (ph -> ch)   =>   |- ((ps /\ th /\ ph) -> ch)
Theorem	3adantl1 803	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ta /\ ph /\ ps) /\ ch) -> th)
Theorem	3adantl2 804	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ta /\ ps) /\ ch) -> th)
Theorem	3adantl3 805	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ps /\ ta) /\ ch) -> th)
Theorem	3adantr1 806	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ta /\ ps /\ ch)) -> th)
Theorem	3adantr2 807	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3adantr3 808	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3ad2antl1 809	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ph /\ ps /\ ta) /\ ch) -> th)
Theorem	3ad2antl2 810	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ps /\ ph /\ ta) /\ ch) -> th)
Theorem	3ad2antl3 811	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ps /\ ta /\ ph) /\ ch) -> th)
Theorem	3ad2antr1 812	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ch /\ ps /\ ta)) -> th)
Theorem	3ad2antr2 813	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3ad2antr3 814	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3mix1 815	Introduction in triple disjunction.
|- (ph -> (ph \/ ps \/ ch))
Theorem	3mix2 816	Introduction in triple disjunction.
|- (ph -> (ps \/ ph \/ ch))
Theorem	3mix3 817	Introduction in triple disjunction.
|- (ph -> (ps \/ ch \/ ph))
Theorem	3pm3.2i 818	Infer conjunction of premises.
|- ph   &   |- ps   &   |- ch   =>   |- (ph /\ ps /\ ch)
Theorem	3jca 819	Join consequents with conjunction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   =>   |- (ph -> (ps /\ ch /\ th))
Theorem	3jcad 820	Deduction conjoining the consequents of three implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (ps -> th))   &   |- (ph -> (ps -> ta))   =>   |- (ph -> (ps -> (ch /\ th /\ ta)))
Theorem	3anim123i 821	Join antecedents and consequents with conjunction.
|- (ph -> ps)   &   |- (ch -> th)   &   |- (ta -> et)   =>   |- ((ph /\ ch /\ ta) -> (ps /\ th /\ et))
Theorem	3anbi123i 822	Join 3 biconditionals with conjunction.
|- (ph <-> ps)   &   |- (ch <-> th)   &   |- (ta <-> et)   =>   |- ((ph /\ ch /\ ta) <-> (ps /\ th /\ et))
Theorem	3orbi123i 823	Join 3 biconditionals with disjunction.
|- (ph <-> ps)   &   |- (ch <-> th)   &   |- (ta <-> et)   =>   |- ((ph \/ ch \/ ta) <-> (ps \/ th \/ et))
Theorem	3anbi1i 824	Inference adding two conjuncts to each side of a biconditional.
|- (ph <-> ps)   =>   |- ((ph /\ ch /\ th) <-> (ps /\ ch /\ th))
Theorem	3anbi2i 825	Inference adding two conjuncts to each side of a biconditional.
|- (ph <-> ps)   =>   |- ((ch /\ ph /\ th) <-> (ch /\ ps /\ th))
Theorem	3anbi3i 826	Inference adding two conjuncts to each side of a biconditional.
|- (ph <-> ps)   =>   |- ((ch /\ th /\ ph) <-> (ch /\ th /\ ps))
Theorem	3imp 827	Importation inference.
|- (ph -> (ps -> (ch -> th)))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impa 828	Importation from double to triple conjunction.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impb 829	Importation from double to triple conjunction.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impia 830	Importation to triple conjunction.
|- ((ph /\ ps) -> (ch -> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impib 831	Importation to triple conjunction.
|- (ph -> ((ps /\ ch) -> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3exp 832	Exportation inference.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> (ps -> (ch -> th)))
Theorem	3expa 833	Exportation from triple to double conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (((ph /\ ps) /\ ch) -> th)
Theorem	3expb 834	Exportation from triple to double conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch)) -> th)
Theorem	3expia 835	Exportation from triple conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> (ch -> th))
Theorem	3expib 836	Exportation from triple conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> ((ps /\ ch) -> th))
Theorem	3com12 837	Commutation in antecedent. Swap 1st and 3rd.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ph /\ ch) -> th)
Theorem	3com13 838	Commutation in antecedent. Swap 1st and 3rd.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ch /\ ps /\ ph) -> th)
Theorem	3com23 839	Commutation in antecedent. Swap 2nd and 3rd.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ch /\ ps) -> th)
Theorem	3coml 840	Commutation in antecedent. Rotate left.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ch /\ ph) -> th)
Theorem	3comr 841	Commutation in antecedent. Rotate right.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ch /\ ph /\ ps) -> th)
Theorem	3adant3r1 842	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ta /\ ps /\ ch)) -> th)
Theorem	3adant3r2 843	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3adant3r3 844	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3an1rs 845	Swap conjuncts.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ps /\ th) /\ ch) -> ta)
Theorem	3imp1 846	Importation to left triple conjunction.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (((ph /\ ps /\ ch) /\ th) -> ta)
Theorem	3impd 847	Importation deduction for triple conjunction.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (ph -> ((ps /\ ch /\ th) -> ta))
Theorem	3imp2 848	Importation to right triple conjunction.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- ((ph /\ (ps /\ ch /\ th)) -> ta)
Theorem	3exp1 849	Exportation from left triple conjunction.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	3expd 850	Exportation deduction for triple conjunction.
|- (ph -> ((ps /\ ch /\ th) -> ta))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	3exp2 851	Exportation from right triple conjunction.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	3adant1l 852	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (((ta /\ ph) /\ ps /\ ch) -> th)
Theorem	3adant1r 853	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (((ph /\ ta) /\ ps /\ ch) -> th)
Theorem	3adant2l 854	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ta /\ ps) /\ ch) -> th)
Theorem	3adant2r 855	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ta) /\ ch) -> th)
Theorem	3adant3l 856	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |-