Theorem elequ1 2207

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 2205	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 2205	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1756	. 2 |- ( x = y -> ( y e. z -> x e. z ) )
4	1, 3	impbid 129	1 |- ( x = y -> ( x e. z <-> y e. z ) )

Syntax hints:   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1498  ax-ie2 1543  ax-8 1553  ax-17 1575  ax-i9 1579  ax-13 2205

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cleljust  2209  elsb1  2210  dveel1  2212  nalset  4240  zfpow  4288  mss  4342  zfun  4555  pw2f1odclem  7087  2omap  7269  ctssdc  7404  acfun  7514  ccfunen  7578  hashfibclem  11206  bj-nalset  16665  bj-nnelirr  16723  pw1map  16769  nninfsellemqall  16793  nninfomni  16797