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Theorem cbvral3vw 3411
 Description: Change bound variables of triple restricted universal quantification, using implicit substitution. Version of cbvral3v 3414 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 10-May-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvral3vw.1 (𝑥 = 𝑤 → (𝜑𝜒))
cbvral3vw.2 (𝑦 = 𝑣 → (𝜒𝜃))
cbvral3vw.3 (𝑧 = 𝑢 → (𝜃𝜓))
Assertion
Ref Expression
cbvral3vw (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
Distinct variable groups:   𝜑,𝑤   𝜓,𝑧   𝜒,𝑥   𝜒,𝑣   𝜃,𝑦   𝜃,𝑢   𝑥,𝐴   𝑧,𝑢   𝑤,𝐴   𝑥,𝑦,𝐵,𝑤   𝑣,𝐵   𝑥,𝑧,𝐶,𝑦,𝑤   𝑧,𝑣,𝐶   𝑢,𝐶   𝑦,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑣,𝑢)   𝜓(𝑥,𝑦,𝑤,𝑣,𝑢)   𝜒(𝑦,𝑧,𝑤,𝑢)   𝜃(𝑥,𝑧,𝑤,𝑣)   𝐴(𝑦,𝑧,𝑣,𝑢)   𝐵(𝑧,𝑢)

Proof of Theorem cbvral3vw
StepHypRef Expression
1 cbvral3vw.1 . . . 4 (𝑥 = 𝑤 → (𝜑𝜒))
212ralbidv 3164 . . 3 (𝑥 = 𝑤 → (∀𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑦𝐵𝑧𝐶 𝜒))
32cbvralvw 3397 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑦𝐵𝑧𝐶 𝜒)
4 cbvral3vw.2 . . . 4 (𝑦 = 𝑣 → (𝜒𝜃))
5 cbvral3vw.3 . . . 4 (𝑧 = 𝑢 → (𝜃𝜓))
64, 5cbvral2vw 3409 . . 3 (∀𝑦𝐵𝑧𝐶 𝜒 ↔ ∀𝑣𝐵𝑢𝐶 𝜓)
76ralbii 3133 . 2 (∀𝑤𝐴𝑦𝐵𝑧𝐶 𝜒 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
83, 7bitri 278 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wral 3106 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clel 2870  df-ral 3111 This theorem is referenced by:  latdisd  17812  dffltz  39786
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