Mathbox for Giovanni Mascellani < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rabeq12f Structured version   Visualization version   GIF version

Theorem rabeq12f 33135
 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 3097 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ {𝑥𝐴𝜑} = {𝑥𝐴𝜓})
21biimpi 205 . 2 (∀𝑥𝐴 (𝜑𝜓) → {𝑥𝐴𝜑} = {𝑥𝐴𝜓})
3 rabeq12f.1 . . 3 𝑥𝐴
4 rabeq12f.2 . . 3 𝑥𝐵
53, 4rabeqf 3165 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
62, 5sylan9eqr 2666 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → {𝑥𝐴𝜑} = {𝑥𝐵𝜓})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnfc 2738  ∀wral 2896  {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-ral 2901  df-rab 2905 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator