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

Theorem raleqbid 3070
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 3054 . . 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 2898 . 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 1379   F/wnf 1599   F/_wnfc 2615   A.wral 2814
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-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator