Axiom ax-11 1397

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 1766). 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.

Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1708, ax11v2 1701 and ax-11o 1704. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Detailed syntax breakdown of Axiom ax-11

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar x
2		vy	. . 3 setvar y
3	1, 2	weq 1392	. 2 wff x = y
4		wph	. . . 4 wff ph
5	4, 2	wal 1241	. . 3 wff A. y ph
6	3, 4	wi 4	. . . 4 wff ( x = y -> ph )
7	6, 1	wal 1241	. . 3 wff A. x ( x = y -> ph )
8	5, 7	wi 4	. 2 wff ( A. y ph -> A. x ( x = y -> ph ) )
9	3, 8	wi 4	1 wff ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )

This axiom is referenced by:  ax10o  1603  equs5a  1675  sbcof2  1691  ax11o  1703  ax11v  1708