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Theorem raleqd 41770
Description: Equality deduction for restricted universal quantifier. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
raleqd.a 𝑥𝐴
raleqd.b 𝑥𝐵
raleqd.e (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
raleqd (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))

Proof of Theorem raleqd
StepHypRef Expression
1 raleqd.e . 2 (𝜑𝐴 = 𝐵)
2 raleqd.a . . 3 𝑥𝐴
3 raleqd.b . . 3 𝑥𝐵
42, 3raleqf 3353 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
51, 4syl 17 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wnfc 2939  wral 3109
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
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 2794  df-clel 2873  df-nfc 2941  df-ral 3114
This theorem is referenced by:  allbutfiinf  42054
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