Theorem elequ1 2182

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 2180	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 2180	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1732	. 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 1473  ax-ie2 1518  ax-8 1528  ax-17 1550  ax-i9 1554  ax-13 2180

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cleljust  2184  elsb1  2185  dveel1  2187  nalset  4190  zfpow  4235  mss  4288  zfun  4499  pw2f1odclem  6956  ctssdc  7241  acfun  7350  ccfunen  7411  bj-nalset  16030  bj-nnelirr  16088  2omap  16132  pw1map  16134  nninfsellemqall  16154  nninfomni  16158