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Theorem raleqf 3343
Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See raleq 3316 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 2910 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2816 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54imbi1d 341 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5albid 2207 . 2 (𝐴 = 𝐵 → (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵𝜑)))
7 df-ral 3056 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
8 df-ral 3056 . 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 1531   = wceq 1533  wcel 2098  wnfc 2877  wral 3055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056
This theorem is referenced by:  rexeqf  3344  raleqbid  3346  dfon2lem3  35290  indexa  37112  ralbi12f  37539  iineq12f  37543  ac6s6f  37552  raleqd  44382  stoweidlem28  45297  stoweidlem52  45321  fourierdlem31  45407  fourierdlem68  45443  fourierdlem103  45478  fourierdlem104  45479
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