Theorem elequ1 1600

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 1404	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 1404	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1594	. 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 1338  ax-ie2 1383  ax-8 1395  ax-13 1404  ax-17 1419  ax-i9 1423

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  cleljust  1813  elsb3  1852  dveel1  1896  nalset  3887  zfpow  3928  mss  3962  zfun  4171  bj-nalset  10015  bj-nnelirr  10078