Theorem elequ1 2168

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 2166	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 2166	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1719	. 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 1460  ax-ie2 1505  ax-8 1515  ax-17 1537  ax-i9 1541  ax-13 2166

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cleljust  2170  elsb1  2171  dveel1  2173  nalset  4159  zfpow  4204  mss  4255  zfun  4465  pw2f1odclem  6890  ctssdc  7172  acfun  7267  ccfunen  7324  bj-nalset  15387  bj-nnelirr  15445  nninfsellemqall  15505  nninfomni  15509