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Theorem rabeqif 38320
 Description: Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
rabeqif.1 𝑥𝐴
rabeqif.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 rabeqif.1 . . 3 𝑥𝐴
3 rabeqif.2 . . 3 𝑥𝐵
42, 3rabeqf 3165 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
51, 4ax-mp 5 1 {𝑥𝐴𝜑} = {𝑥𝐵𝜑}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  Ⅎwnfc 2738  {crab 2900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905 This theorem is referenced by:  rabeqi  38332
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