Axiom ax-11 1180

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 1251). 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 1202 ("o" for "old"), which was replaced with this shorter ax-11 1180 in Jan. 2007.

Juha Arpiainen proved the independence of this axiom (in the form of the older axiom ax-11o 1202) 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 1202 (from which the ax-11 1180 instance follows by theorem ax11 1203.) The proof is by induction on formula length, using ax11eq 1340 and ax11el 1341 for the basis steps and ax11indn 1343, ax11indi 1344, and ax11inda 1348 for the induction steps.

See also ax11v 1249 and ax11v2 1199 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 1098	. . 3 class x
3		vy	. . . 4 set y
4	3	cv 1098	. . 3 class y
5	2, 4	wceq 1099	. 2 wff x = y
6		wph	. . . 4 wff ph
7	6, 3	wal 950	. . 3 wff A.yph
8	5, 6	wi 3	. . . 4 wff (x = y -> ph)
9	8, 1	wal 950	. . 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:  equs5a 1181  equs5e 1182  ax11o 1201

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