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Theorem rabeqif 2649
Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2648. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
rabeqf.1 𝑥𝐴
rabeqf.2 𝑥𝐵
rabeqif.3 𝐴 = 𝐵
Assertion
Ref Expression
rabeqif {𝑥𝐴𝜑} = {𝑥𝐵𝜑}

Proof of Theorem rabeqif
StepHypRef Expression
1 rabeqif.3 . 2 𝐴 = 𝐵
2 rabeqf.1 . . 3 𝑥𝐴
3 rabeqf.2 . . 3 𝑥𝐵
42, 3rabeqf 2648 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
51, 4ax-mp 5 1 {𝑥𝐴𝜑} = {𝑥𝐵𝜑}
Colors of variables: wff set class
Syntax hints:   = wceq 1314  wnfc 2243  {crab 2395
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400
This theorem is referenced by:  rabeqi  2651
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