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Theorem raleqbid 3063
Description: Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypotheses
Ref Expression
raleqbid.0  |-  F/ x ph
raleqbid.1  |-  F/_ x A
raleqbid.2  |-  F/_ x B
raleqbid.3  |-  ( ph  ->  A  =  B )
raleqbid.4  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
raleqbid  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
)

Proof of Theorem raleqbid
StepHypRef Expression
1 raleqbid.3 . . 3  |-  ( ph  ->  A  =  B )
2 raleqbid.1 . . . 4  |-  F/_ x A
3 raleqbid.2 . . . 4  |-  F/_ x B
42, 3raleqf 3047 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  ps 
<-> 
A. x  e.  B  ps ) )
51, 4syl 16 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ps )
)
6 raleqbid.0 . . 3  |-  F/ x ph
7 raleqbid.4 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
86, 7ralbid 2888 . 2  |-  ( ph  ->  ( A. x  e.  B  ps  <->  A. x  e.  B  ch )
)
95, 8bitrd 253 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398   F/wnf 1621   F/_wnfc 2602   A.wral 2804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809
This theorem is referenced by: (None)
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