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Theorem rabeqif 2610
Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2609. (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 2609 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
51, 4ax-mp 7 1 {𝑥𝐴𝜑} = {𝑥𝐵𝜑}
Colors of variables: wff set class
Syntax hints:   = wceq 1289  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:  rabeqi  2612
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