Theorem elequ1 1713

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 1712	. 2 |- (x = y -> (x e. z -> y e. z))
2		ax-13 1712	. . 3 |- (y = x -> (y e. z -> x e. z))
3	2	equcoms 1681	. 2 |- (x = y -> (y e. z -> x e. z))
4	1, 3	impbid 183	1 |- (x = y -> (x e. z <-> y e. z))

Syntax hints:   -> wi 4   <-> wb 176

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  ax11wdemo  1723  cleljust  2014  cleljustALT  2015  dveel1  2019  ax15  2021  elsb3  2103  ax11el  2194  ncfinlower  4483  tfinnn  4534  tz6.12-2  5346