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Theorem rexeqbii 2894
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 2541 . . 3  |-  ( x  e.  A  <->  x  e.  B )
3 rexeqbii.2 . . 3  |-  ( ps  <->  ch )
42, 3anbi12i 711 . 2  |-  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch )
)
54rexbii2 2879 1  |-  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    = wceq 1452    e. wcel 1904   E.wrex 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-cleq 2464  df-clel 2467  df-rex 2762
This theorem is referenced by:  bnj882  29809
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