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Theorem rexeqbii 2856
Description: Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
rexeqbii.1  |-  A  =  B
rexeqbii.2  |-  ( ps  <->  ch )
Assertion
Ref Expression
rexeqbii  |-  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )

Proof of Theorem rexeqbii
StepHypRef Expression
1 rexeqbii.1 . . . 4  |-  A  =  B
21eleq2i 2529 . . 3  |-  ( x  e.  A  <->  x  e.  B )
3 rexeqbii.2 . . 3  |-  ( ps  <->  ch )
42, 3anbi12i 697 . 2  |-  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch )
)
54rexbii2 2851 1  |-  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   E.wrex 2796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-cleq 2443  df-clel 2446  df-rex 2801
This theorem is referenced by:  bnj882  32221
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