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Theorem raleqbidva 2519
Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
Hypotheses
Ref Expression
raleqbidva.1  |-  ( ph  ->  A  =  B )
raleqbidva.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
raleqbidva  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
)
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hints:    ps( x)    ch( x)

Proof of Theorem raleqbidva
StepHypRef Expression
1 raleqbidva.2 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
21ralbidva 2322 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  A  ch )
)
3 raleqbidva.1 . . 3  |-  ( ph  ->  A  =  B )
43raleqdv 2511 . 2  |-  ( ph  ->  ( A. x  e.  A  ch  <->  A. x  e.  B  ch )
)
52, 4bitrd 177 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311
This theorem is referenced by: (None)
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