Theorem elequ1 2140

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 2138	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 2138	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1696	. 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 1437  ax-ie2 1482  ax-8 1492  ax-17 1514  ax-i9 1518  ax-13 2138

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  cleljust  2142  elsb1  2143  dveel1  2145  nalset  4112  zfpow  4154  mss  4204  zfun  4412  ctssdc  7078  acfun  7163  ccfunen  7205  bj-nalset  13777  bj-nnelirr  13835  nninfsellemqall  13895  nninfomni  13899