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

Theorem raleqbidva 3033
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 2841 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  A  ch )
)
3 raleqbidva.1 . . 3  |-  ( ph  ->  A  =  B )
43raleqdv 3023 . 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 369    = wceq 1370    e. wcel 1758   A.wral 2796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801
This theorem is referenced by:  swrdspsleq  12455  catpropd  14762  cidpropd  14763  funcpropd  14924  fullpropd  14944  natpropd  15000  gsumpropd2lem  15619  istrkgc  23043  istrkgb  23044  istrkgcb  23045  istrkge  23046  iscgrg  23096  isperp  23243  rngurd  26396  clwlkisclwwlk  30594
  Copyright terms: Public domain W3C validator