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Theorem raleqbii 2912
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 2545 . . 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 2896 1  |-  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   A.wral 2817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-cleq 2459  df-clel 2462  df-ral 2822
This theorem is referenced by:  ply1coe  18207  ordtbaslem  19557  iscusp2  20673  elghom  25188  wfrlem5  29274  frrlem5  29318  iscrngo2  30322  tendoset  35956
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