Axiom ax-11 1370

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 1739). 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 1681, ax11v2 1674 and ax-11o 1677. (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 1365	. 2 wff x = y
4		wph	. . . 4 wff ph
5	4, 2	wal 1221	. . 3 wff A.yph
6	3, 4	wi 4	. . . 4 wff (x = y -> ph)
7	6, 1	wal 1221	. . 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  1576  equs5a  1648  sbcof2  1664  ax11o  1676  ax11v  1681