Theorem elequ1 2206

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 2204	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 2204	. . 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 1497  ax-ie2 1542  ax-8 1552  ax-17 1574  ax-i9 1578  ax-13 2204

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cleljust  2208  elsb1  2209  dveel1  2211  nalset  4219  zfpow  4265  mss  4318  zfun  4531  pw2f1odclem  7019  ctssdc  7311  acfun  7421  ccfunen  7482  bj-nalset  16490  bj-nnelirr  16548  2omap  16594  pw1map  16596  nninfsellemqall  16617  nninfomni  16621