MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  raleqbidva Structured version   Unicode version

Theorem raleqbidva 2910
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 2713 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  A  ch )
)
3 raleqbidva.1 . . 3  |-  ( ph  ->  A  =  B )
43raleqdv 2902 . 2  |-  ( ph  ->  ( A. x  e.  A  ch  <->  A. x  e.  B  ch )
)
52, 4bitrd 245 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697
This theorem is referenced by:  catpropd  13927  cidpropd  13928  funcpropd  14089  fullpropd  14109  natpropd  14165  gsumpropd2lem  24212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702
  Copyright terms: Public domain W3C validator