Theorem elequ1 2213

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 2211	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 2211	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1760	. 2 |- ( x = y -> ( y e. z -> x e. z ) )
4	1, 3	impbid 129	1 |- ( x = y -> ( x e. z <-> y e. z ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1502  ax-ie2 1547  ax-8 1557  ax-17 1579  ax-i9 1583  ax-13 2211

This proof depends on definitions:  df-bi 117

This theorem is used by:  cleljust  2215  elsb1  2216  dveel1  2218  nalset  4263  zfpow  4312  mss  4366  zfun  4579  pw2f1odclem  7134  2omap  7318  ctssdc  7453  acfun  7563  ccfunen  7630  hashfibclem  11282  bj-nalset  16921  bj-nnelirr  16979  pw1map  17025  nninfsellemqall  17058  nninfomni  17062