Theorem elequ1 2204

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 2202	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 2202	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1754	. 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 1495  ax-ie2 1540  ax-8 1550  ax-17 1572  ax-i9 1576  ax-13 2202

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cleljust  2206  elsb1  2207  dveel1  2209  nalset  4213  zfpow  4258  mss  4311  zfun  4524  pw2f1odclem  6991  ctssdc  7276  acfun  7385  ccfunen  7446  bj-nalset  16216  bj-nnelirr  16274  2omap  16318  pw1map  16320  nninfsellemqall  16340  nninfomni  16344