Theorem elequ1 1640

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 1444	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 1444	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1634	. 2 |- ( x = y -> ( y e. z -> x e. z ) )
4	1, 3	impbid 127	1 |- ( x = y -> ( x e. z <-> y e. z ) )

Syntax hints:   -> wi 4   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1378  ax-ie2 1423  ax-8 1435  ax-13 1444  ax-17 1459  ax-i9 1463

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  cleljust  1854  elsb3  1893  dveel1  1937  nalset  3908  zfpow  3949  mss  3981  zfun  4189  bj-nalset  10686  bj-nnelirr  10748