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Theorem raleqbii 2489
Description: Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
raleqbii.1  |-  A  =  B
raleqbii.2  |-  ( ps  <->  ch )
Assertion
Ref Expression
raleqbii  |-  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )

Proof of Theorem raleqbii
StepHypRef Expression
1 raleqbii.1 . . . 4  |-  A  =  B
21eleq2i 2244 . . 3  |-  ( x  e.  A  <->  x  e.  B )
3 raleqbii.2 . . 3  |-  ( ps  <->  ch )
42, 3imbi12i 239 . 2  |-  ( ( x  e.  A  ->  ps )  <->  ( x  e.  B  ->  ch )
)
54ralbii2 2487 1  |-  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455
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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-ral 2460
This theorem is referenced by: (None)
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