Axiom ax-11 1374

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 1744). 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 1686, ax11v2 1679 and ax-11o 1682. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ax-11	|- (x = y -> (A.yph -> 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 1369	. 2 wff x = y
4		wph	. . . 4 wff ph
5	4, 2	wal 1224	. . 3 wff A.yph
6	3, 4	wi 4	. . . 4 wff (x = y -> ph)
7	6, 1	wal 1224	. . 3 wff A.x(x = y -> ph)
8	5, 7	wi 4	. 2 wff (A.yph -> A.x(x = y -> ph))
9	3, 8	wi 4	1 wff (x = y -> (A.yph -> A.x(x = y -> ph)))

This axiom is referenced by:  ax10o  1581  equs5a  1653  sbcof2  1669  ax11o  1681  ax11v  1686