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Theorem ralbi12f 35878
 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 3099 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓))
2 ralbi12f.1 . . 3 𝑥𝐴
3 ralbi12f.2 . . 3 𝑥𝐵
42, 3raleqf 3315 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
51, 4sylan9bbr 514 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnfc 2899  ∀wral 3070 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 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075 This theorem is referenced by: (None)
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