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Theorem rabeq12f 35614
 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 3336 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ {𝑥𝐴𝜑} = {𝑥𝐴𝜓})
21biimpi 219 . 2 (∀𝑥𝐴 (𝜑𝜓) → {𝑥𝐴𝜑} = {𝑥𝐴𝜓})
3 rabeq12f.1 . . 3 𝑥𝐴
4 rabeq12f.2 . . 3 𝑥𝐵
53, 4rabeqf 3428 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
62, 5sylan9eqr 2855 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → {𝑥𝐴𝜑} = {𝑥𝐵𝜓})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnfc 2936  ∀wral 3106  {crab 3110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115 This theorem is referenced by: (None)
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