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Theorem raleqf 3111
 Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1 𝑥𝐴
raleq1f.2 𝑥𝐵
Assertion
Ref Expression
raleqf (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))

Proof of Theorem raleqf
StepHypRef Expression
1 raleq1f.1 . . . 4 𝑥𝐴
2 raleq1f.2 . . . 4 𝑥𝐵
31, 2nfeq 2762 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2677 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54imbi1d 330 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5albid 2077 . 2 (𝐴 = 𝐵 → (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵𝜑)))
7 df-ral 2901 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
8 df-ral 2901 . 2 (∀𝑥𝐵 𝜑 ↔ ∀𝑥(𝑥𝐵𝜑))
96, 7, 83bitr4g 302 1 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901 This theorem is referenced by:  raleq  3115  raleqbid  3127  dfon2lem3  30934  indexa  32698  ralbi12f  33139  iineq12f  33143  ac6s6f  33151  stoweidlem28  38921  stoweidlem52  38945  fourierdlem31  39031  fourierdlem68  39067  fourierdlem103  39102  fourierdlem104  39103
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