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Theorem raleqbii 2766
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 2507 . . 3  |-  ( x  e.  A  <->  x  e.  B )
3 raleqbii.2 . . 3  |-  ( ps  <->  ch )
42, 3imbi12i 326 . 2  |-  ( ( x  e.  A  ->  ps )  <->  ( x  e.  B  ->  ch )
)
54ralbii2 2764 1  |-  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756   A.wral 2736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-cleq 2436  df-clel 2439  df-ral 2741
This theorem is referenced by:  ply1coe  17768  ordtbaslem  18814  iscusp2  19899  elghom  23872  wfrlem5  27750  frrlem5  27794  iscrngo2  28824  tendoset  34499
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