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Theorem rabeq12f 33936
Description: Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Hypotheses
Ref Expression
rabeq12f.1 𝑥𝐴
rabeq12f.2 𝑥𝐵
Assertion
Ref Expression
rabeq12f ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → {𝑥𝐴𝜑} = {𝑥𝐵𝜓})

Proof of Theorem rabeq12f
StepHypRef Expression
1 rabbi 3115 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ {𝑥𝐴𝜑} = {𝑥𝐴𝜓})
21biimpi 206 . 2 (∀𝑥𝐴 (𝜑𝜓) → {𝑥𝐴𝜑} = {𝑥𝐴𝜓})
3 rabeq12f.1 . . 3 𝑥𝐴
4 rabeq12f.2 . . 3 𝑥𝐵
53, 4rabeqf 3185 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
62, 5sylan9eqr 2676 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → {𝑥𝐴𝜑} = {𝑥𝐵𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wnfc 2749  wral 2909  {crab 2913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918
This theorem is referenced by: (None)
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