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Theorem rabeqif 2651
Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2650. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
rabeqf.1  |-  F/_ x A
rabeqf.2  |-  F/_ x B
rabeqif.3  |-  A  =  B
Assertion
Ref Expression
rabeqif  |-  { x  e.  A  |  ph }  =  { x  e.  B  |  ph }

Proof of Theorem rabeqif
StepHypRef Expression
1 rabeqif.3 . 2  |-  A  =  B
2 rabeqf.1 . . 3  |-  F/_ x A
3 rabeqf.2 . . 3  |-  F/_ x B
42, 3rabeqf 2650 . 2  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
51, 4ax-mp 5 1  |-  { x  e.  A  |  ph }  =  { x  e.  B  |  ph }
Colors of variables: wff set class
Syntax hints:    = wceq 1316   F/_wnfc 2245   {crab 2397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402
This theorem is referenced by:  rabeqi  2653
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