Theorem elequ1 2145

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 2143	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 2143	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1701	. 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 1442  ax-ie2 1487  ax-8 1497  ax-17 1519  ax-i9 1523  ax-13 2143

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  cleljust  2147  elsb1  2148  dveel1  2150  nalset  4119  zfpow  4161  mss  4211  zfun  4419  ctssdc  7090  acfun  7184  ccfunen  7226  bj-nalset  13930  bj-nnelirr  13988  nninfsellemqall  14048  nninfomni  14052