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Theorem raleqf 3353
Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See raleq 3323 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 2919 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2830 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54imbi1d 341 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5albid 2222 . 2 (𝐴 = 𝐵 → (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵𝜑)))
7 df-ral 3062 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
8 df-ral 3062 . 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 206  wal 1538   = wceq 1540  wcel 2108  wnfc 2890  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062
This theorem is referenced by:  rexeqf  3354  raleqbid  3356  dfon2lem3  35786  indexa  37740  ralbi12f  38167  iineq12f  38171  ac6s6f  38180  raleqd  45142  stoweidlem28  46043  stoweidlem52  46067  fourierdlem31  46153  fourierdlem68  46189  fourierdlem103  46224  fourierdlem104  46225
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