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Theorem raleqbidva 3067
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 2890 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  A  ch )
)
3 raleqbidva.1 . . 3  |-  ( ph  ->  A  =  B )
43raleqdv 3057 . 2  |-  ( ph  ->  ( A. x  e.  A  ch  <->  A. x  e.  B  ch )
)
52, 4bitrd 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    /\ wa 367    = wceq 1398    e. wcel 1823   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:  catpropd  15200  cidpropd  15201  funcpropd  15391  fullpropd  15411  natpropd  15467  gsumpropd2lem  16102  istrkgc  24052  istrkgb  24053  istrkgcb  24054  istrkge  24055  iscgrg  24108  isperp  24293  clwlkisclwwlk  24994  rngurd  28016
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