Theorem elequ2 2153

Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
elequ2	|- ( x = y -> ( z e. x <-> z e. y ) )

Proof of Theorem elequ2

Step	Hyp	Ref	Expression
1		ax-14 2151	. 2 |- ( x = y -> ( z e. x -> z e. y ) )
2		ax-14 2151	. . 3 |- ( y = x -> ( z e. y -> z e. x ) )
3	2	equcoms 1708	. 2 |- ( x = y -> ( z e. y -> z e. x ) )
4	1, 3	impbid 129	1 |- ( x = y -> ( z e. x <-> z e. y ) )

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 1449  ax-ie2 1494  ax-8 1504  ax-17 1526  ax-i9 1530  ax-14 2151

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  elsb2  2156  dveel2  2158  axext3  2160  axext4  2161  bm1.1  2162  eleq2w  2239  bm1.3ii  4125  nalset  4134  zfun  4435  fv3  5539  tfrlemisucaccv  6326  tfr1onlemsucaccv  6342  tfrcllemsucaccv  6355  acfun  7206  ccfunen  7263  cc1  7264  bdsepnft  14642  bdsepnfALT  14644  bdbm1.3ii  14646  bj-nalset  14650  bj-nnelirr  14708  nninfalllem1  14760  nninfsellemeq  14766  nninfsellemqall  14767  nninfsellemeqinf  14768  nninfomni  14771