Theorem elequ1 2213

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 2211	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 2211	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1760	. 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 1502  ax-ie2 1547  ax-8 1557  ax-17 1579  ax-i9 1583  ax-13 2211

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cleljust  2215  elsb1  2216  dveel1  2218  nalset  4258  zfpow  4307  mss  4361  zfun  4574  pw2f1odclem  7124  2omap  7308  ctssdc  7443  acfun  7553  ccfunen  7620  hashfibclem  11260  bj-nalset  16835  bj-nnelirr  16893  pw1map  16939  nninfsellemqall  16963  nninfomni  16967