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Theorem raleqbii 2737
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 2502 . . 3  |-  ( x  e.  A  <->  x  e.  B )
3 raleqbii.2 . . 3  |-  ( ps  <->  ch )
42, 3imbi12i 318 . 2  |-  ( ( x  e.  A  ->  ps )  <->  ( x  e.  B  ->  ch )
)
54ralbii2 2735 1  |-  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   A.wral 2707
This theorem is referenced by:  ordtbaslem  17254  iscusp2  18334  elghom  21953  wfrlem5  25544  frrlem5  25588  iscrngo2  26610  tendoset  31558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2431  df-clel 2434  df-ral 2712
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