Axiom ax-11 969

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 1269). 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 1220 ("o" for "old"), which was replaced with this shorter ax-11 969 in Jan. 2007.

Juha Arpiainen proved the independence of this axiom (in the form of the older axiom ax-11o 1220) 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 1220 (from which the ax-11 969 instance follows by theorem ax11 1221.) The proof is by induction on formula length, using ax11eq 1365 and ax11el 1366 for the basis steps and ax11indn 1368, ax11indi 1369, and ax11inda 1373 for the induction steps.

See also ax11v 1267 and ax11v2 1217 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 957	. . 3 class x
3		vy	. . . 4 set y
4	3	cv 957	. . 3 class y
5	2, 4	wceq 958	. 2 wff x = y
6		wph	. . . 4 wff ph
7	6, 3	wal 956	. . 3 wff A.yph
8	5, 6	wi 3	. . . 4 wff (x = y -> ph)
9	8, 1	wal 956	. . 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 974  ax10o 1141  equs5a 1199  equs5e 1200  ax11o 1219

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