Axiom ax-11 1528

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 1909). 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 1849, ax11v2 1842 and ax-11o 1845. (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 1525	. 2 wff x = y
4		wph	. . . 4 wff ph
5	4, 2	wal 1370	. . 3 wff A. y ph
6	3, 4	wi 4	. . . 4 wff ( x = y -> ph )
7	6, 1	wal 1370	. . 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  1737  equs5a  1816  sbcof2  1832  ax11o  1844  ax11v  1849