Theorem elequ1 2180

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 2178	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 2178	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1731	. 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 1472  ax-ie2 1517  ax-8 1527  ax-17 1549  ax-i9 1553  ax-13 2178

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cleljust  2182  elsb1  2183  dveel1  2185  nalset  4174  zfpow  4219  mss  4270  zfun  4481  pw2f1odclem  6931  ctssdc  7215  acfun  7319  ccfunen  7376  bj-nalset  15831  bj-nnelirr  15889  2omap  15932  nninfsellemqall  15952  nninfomni  15956