Axiom ax-11 1529

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 1910). 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 1850, ax11v2 1843 and ax-11o 1846. (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 1526	. 2 wff x = y
4		wph	. . . 4 wff ph
5	4, 2	wal 1371	. . 3 wff A. y ph
6	3, 4	wi 4	. . . 4 wff ( x = y -> ph )
7	6, 1	wal 1371	. . 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  1738  equs5a  1817  sbcof2  1833  ax11o  1845  ax11v  1850