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Theorem raleqbid 3330
Description: Equality deduction for restricted universal quantifier. See raleqbidv 3319 for a version based on fewer axioms. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypotheses
Ref Expression
raleqbid.0 𝑥𝜑
raleqbid.1 𝑥𝐴
raleqbid.2 𝑥𝐵
raleqbid.3 (𝜑𝐴 = 𝐵)
raleqbid.4 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
raleqbid (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Proof of Theorem raleqbid
StepHypRef Expression
1 raleqbid.3 . . 3 (𝜑𝐴 = 𝐵)
2 raleqbid.1 . . . 4 𝑥𝐴
3 raleqbid.2 . . . 4 𝑥𝐵
42, 3raleqf 3328 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
51, 4syl 17 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
6 raleqbid.0 . . 3 𝑥𝜑
7 raleqbid.4 . . 3 (𝜑 → (𝜓𝜒))
86, 7ralbid 3256 . 2 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑥𝐵 𝜒))
95, 8bitrd 278 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wnf 1785  wnfc 2887  wral 3064
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065
This theorem is referenced by: (None)
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