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Theorem rabeqif 2700
 Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2699. (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 2699 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
51, 4ax-mp 5 1 {𝑥𝐴𝜑} = {𝑥𝐵𝜑}
 Colors of variables: wff set class Syntax hints:   = wceq 1332  Ⅎwnfc 2283  {crab 2436 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rab 2441 This theorem is referenced by:  rabeqi  2702
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