Theorem elequ1 1690

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 1491	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 1491	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1684	. 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 1425  ax-ie2 1470  ax-8 1482  ax-13 1491  ax-17 1506  ax-i9 1510

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  cleljust  1908  elsb3  1949  dveel1  1993  nalset  4053  zfpow  4094  mss  4143  zfun  4351  ctssdc  6991  acfun  7056  ccfunen  7072  bj-nalset  13082  bj-nnelirr  13140  nninfsellemqall  13200  nninfomni  13204