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Theorem raleqf 3346
Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See raleq 3319 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 2913 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2818 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54imbi1d 341 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5albid 2211 . 2 (𝐴 = 𝐵 → (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵𝜑)))
7 df-ral 3059 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
8 df-ral 3059 . 2 (∀𝑥𝐵 𝜑 ↔ ∀𝑥(𝑥𝐵𝜑))
96, 7, 83bitr4g 314 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 2879  wral 3058
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 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059
This theorem is referenced by:  rexeqf  3347  raleqbid  3349  dfon2lem3  35381  indexa  37206  ralbi12f  37633  iineq12f  37637  ac6s6f  37646  raleqd  44503  stoweidlem28  45416  stoweidlem52  45440  fourierdlem31  45526  fourierdlem68  45562  fourierdlem103  45597  fourierdlem104  45598
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