Axiom ax-11 1555

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 1937). 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 1876, ax11v2 1869 and ax-11o 1872. (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 1552	. 2 wff x = y
4		wph	. . . 4 wff ph
5	4, 2	wal 1396	. . 3 wff A. y ph
6	3, 4	wi 4	. . . 4 wff ( x = y -> ph )
7	6, 1	wal 1396	. . 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  1763  equs5a  1843  sbcof2  1859  ax11o  1871  ax11v  1876