Theorem elequ1 1647

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 1449	. 2 |- ( x = y -> ( x e. z -> y e. z ) )
2		ax-13 1449	. . 3 |- ( y = x -> ( y e. z -> x e. z ) )
3	2	equcoms 1641	. 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 1383  ax-ie2 1428  ax-8 1440  ax-13 1449  ax-17 1464  ax-i9 1468

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  cleljust  1861  elsb3  1900  dveel1  1944  nalset  3961  zfpow  4002  mss  4044  zfun  4252  bj-nalset  11443  bj-nnelirr  11505  nninfsellemqall  11564  nninfomni  11568