Axiom ax-11 965

Description: Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent A.x(x = y -> ph) is a way of expressing "y substituted for x in wff ph" (cf. sb6 1265). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases.

The original version of this axiom, that isn't otherwise used in our development, was ax-11o 1216 ("o" for "old"), which was replaced with this shorter ax-11 965 in Jan. 2007.

Juha Arpiainen proved the independence of this axiom (in the form of the older axiom ax-11o 1216) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

Interestingly, if the wff expression substituted for ph contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-11o 1216 (from which the ax-11 965 instance follows by theorem ax11 1217.) The proof is by induction on formula length, using ax11eq 1361 and ax11el 1362 for the basis steps and ax11indn 1364, ax11indi 1365, and ax11inda 1369 for the induction steps.

See also ax11v 1263 and ax11v2 1213 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

Assertion

Ref	Expression
ax-11	|- (x = y -> (A.yph -> A.x(x = y -> ph)))

Detailed syntax breakdown of Axiom ax-11

Step	Hyp	Ref	Expression
1		vx	. . . 4 set x
2	1	cv 953	. . 3 class x
3		vy	. . . 4 set y
4	3	cv 953	. . 3 class y
5	2, 4	wceq 954	. 2 wff x = y
6		wph	. . . 4 wff ph
7	6, 3	wal 952	. . 3 wff A.yph
8	5, 6	wi 3	. . . 4 wff (x = y -> ph)
9	8, 1	wal 952	. . 3 wff A.x(x = y -> ph)
10	7, 9	wi 3	. 2 wff (A.yph -> A.x(x = y -> ph))
11	5, 10	wi 3	1 wff (x = y -> (A.yph -> A.x(x = y -> ph)))

This axiom is referenced by:  ax4 970  ax10o 1137  equs5a 1195  equs5e 1196  ax11o 1215

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