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Theorem raleqf 3337
Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See raleq 3312 for a version based on fewer axioms. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleqf.1 𝑥𝐴
raleqf.2 𝑥𝐵
Assertion
Ref Expression
raleqf (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))

Proof of Theorem raleqf
StepHypRef Expression
1 raleqf.1 . . . 4 𝑥𝐴
2 raleqf.2 . . . 4 𝑥𝐵
31, 2nfeq 2906 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2815 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54imbi1d 340 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5albid 2211 . 2 (𝐴 = 𝐵 → (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵𝜑)))
7 df-ral 3052 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
8 df-ral 3052 . 2 (∀𝑥𝐵 𝜑 ↔ ∀𝑥(𝑥𝐵𝜑))
96, 7, 83bitr4g 313 1 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532   = wceq 1534  wcel 2099  wnfc 2876  wral 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052
This theorem is referenced by:  rexeqf  3338  raleqbid  3340  dfon2lem3  35611  indexa  37436  ralbi12f  37863  iineq12f  37867  ac6s6f  37876  raleqd  44756  stoweidlem28  45667  stoweidlem52  45691  fourierdlem31  45777  fourierdlem68  45813  fourierdlem103  45848  fourierdlem104  45849
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