Axiom ax-11 1437

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 1807). 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 1748, ax11v2 1741 and ax-11o 1744. (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 1432	. 2 wff x = y
4		wph	. . . 4 wff ph
5	4, 2	wal 1282	. . 3 wff A. y ph
6	3, 4	wi 4	. . . 4 wff ( x = y -> ph )
7	6, 1	wal 1282	. . 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  1643  equs5a  1715  sbcof2  1731  ax11o  1743  ax11v  1748