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

Theorem rabeqbidva 3066
Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
rabeqbidva.1  |-  ( ph  ->  A  =  B )
rabeqbidva.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rabeqbidva  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hints:    ps( x)    ch( x)

Proof of Theorem rabeqbidva
StepHypRef Expression
1 rabeqbidva.2 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
21rabbidva 3061 . 2  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  A  |  ch } )
3 rabeqbidva.1 . . 3  |-  ( ph  ->  A  =  B )
4 rabeq 3064 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  ch }  =  { x  e.  B  |  ch } )
53, 4syl 16 . 2  |-  ( ph  ->  { x  e.  A  |  ch }  =  {
x  e.  B  |  ch } )
62, 5eqtrd 2492 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2799
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-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rab 2804
This theorem is referenced by:  natpropd  14990  gsumpropd2lem  15609  eengv  23362  elntg  23367  domnmsuppn0  30922
  Copyright terms: Public domain W3C validator