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Theorem raleqd 45076
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 3350 . 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 206   = wceq 1536  wnfc 2887  wral 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059
This theorem is referenced by:  allbutfiinf  45369
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