Theorem elequ1 1691

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 1492	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 1492	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1685	. 2 |- ( x = y -> ( y e. z -> x e. z ) )
4	1, 3	impbid 128	1 |- ( x = y -> ( x e. z <-> y e. z ) )

Syntax hints:   -> wi 4   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1426  ax-ie2 1471  ax-8 1483  ax-13 1492  ax-17 1507  ax-i9 1511

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  cleljust  1911  elsb3  1952  dveel1  1996  nalset  4066  zfpow  4107  mss  4156  zfun  4364  ctssdc  7006  acfun  7080  ccfunen  7096  bj-nalset  13264  bj-nnelirr  13322  nninfsellemqall  13386  nninfomni  13390