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Theorem rabeqf 2951
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 2581 . . 3  |-  F/ x  A  =  B
4 eleq2 2499 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 687 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5abbid 2551 . 2  |-  ( A  =  B  ->  { x  |  ( x  e.  A  /\  ph ) }  =  { x  |  ( x  e.  B  /\  ph ) } )
7 df-rab 2716 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
8 df-rab 2716 . 2  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
96, 7, 83eqtr4g 2495 1  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   F/_wnfc 2561   {crab 2711
This theorem is referenced by:  rabeq  2952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716
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