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Theorem rabeqf 2609
Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
Hypotheses
Ref Expression
rabeqf.1  |-  F/_ x A
rabeqf.2  |-  F/_ x B
Assertion
Ref Expression
rabeqf  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )

Proof of Theorem rabeqf
StepHypRef Expression
1 rabeqf.1 . . . 4  |-  F/_ x A
2 rabeqf.2 . . . 4  |-  F/_ x B
31, 2nfeq 2236 . . 3  |-  F/ x  A  =  B
4 eleq2 2151 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 453 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5abbid 2204 . 2  |-  ( A  =  B  ->  { x  |  ( x  e.  A  /\  ph ) }  =  { x  |  ( x  e.  B  /\  ph ) } )
7 df-rab 2368 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
8 df-rab 2368 . 2  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
96, 7, 83eqtr4g 2145 1  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   {cab 2074   F/_wnfc 2215   {crab 2363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368
This theorem is referenced by:  rabeqif  2610  rabeq  2611
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