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Theorem raleqd 44287
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 3348 . 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 205   = wceq 1540  wnfc 2882  wral 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061
This theorem is referenced by:  allbutfiinf  44588
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