Axiom ax-14 1445

Description: Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the e. binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ax-14	|- ( x = y -> ( z e. x -> z e. y ) )

Detailed syntax breakdown of Axiom ax-14

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar x
2		vy	. . 3 setvar y
3	1, 2	weq 1432	. 2 wff x = y
4		vz	. . . 4 setvar z
5	4, 1	wel 1434	. . 3 wff z e. x
6	4, 2	wel 1434	. . 3 wff z e. y
7	5, 6	wi 4	. 2 wff ( z e. x -> z e. y )
8	3, 7	wi 4	1 wff ( x = y -> ( z e. x -> z e. y ) )

This axiom is referenced by:  elequ2  1641  el  3952  fv3  5218