Theorem elequ1 1597

Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
elequ1	|- (x = y -> (x e. z <-> y e. z))

Proof of Theorem elequ1

Step	Hyp	Ref	Expression
1		ax-13 1401	. 2 |- (x = y -> (x e. z -> y e. z))
2		ax-13 1401	. . 3 |- (y = x -> (y e. z -> x e. z))
3	2	equcoms 1591	. 2 |- (x = y -> (y e. z -> x e. z))
4	1, 3	impbid 120	1 |- (x = y -> (x e. z <-> y e. z))

Syntax hints:   -> wi 4   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1335  ax-ie2 1380  ax-8 1392  ax-13 1401  ax-17 1416  ax-i9 1420

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  cleljust  1810  elsb3  1849  dveel1  1893  nalset  3878  zfpow  3919  mss  3953  zfun  4137  bj-nalset  9350  bj-nnelirr  9413