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Theorem ralbi12f 34989
Description: Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Hypotheses
Ref Expression
ralbi12f.1 𝑥𝐴
ralbi12f.2 𝑥𝐵
Assertion
Ref Expression
ralbi12f ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))

Proof of Theorem ralbi12f
StepHypRef Expression
1 ralbi 3134 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓))
2 ralbi12f.1 . . 3 𝑥𝐴
3 ralbi12f.2 . . 3 𝑥𝐵
42, 3raleqf 3357 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
51, 4sylan9bbr 511 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wnfc 2933  wral 3105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110
This theorem is referenced by: (None)
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