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Theorem cbvral3v 2543
 Description: Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
Hypotheses
Ref Expression
cbvral3v.1 (𝑥 = 𝑤 → (𝜑𝜒))
cbvral3v.2 (𝑦 = 𝑣 → (𝜒𝜃))
cbvral3v.3 (𝑧 = 𝑢 → (𝜃𝜓))
Assertion
Ref Expression
cbvral3v (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
Distinct variable groups:   𝜑,𝑤   𝜓,𝑧   𝜒,𝑥   𝜒,𝑣   𝑦,𝑢,𝜃   𝑥,𝐴   𝑤,𝐴   𝑥,𝑦,𝐵   𝑦,𝑤,𝐵   𝑣,𝐵   𝑥,𝑧,𝐶,𝑦   𝑧,𝑤,𝐶   𝑧,𝑣,𝐶   𝑢,𝐶
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 cbvral3v
StepHypRef Expression
1 cbvral3v.1 . . . 4 (𝑥 = 𝑤 → (𝜑𝜒))
212ralbidv 2348 . . 3 (𝑥 = 𝑤 → (∀𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑦𝐵𝑧𝐶 𝜒))
32cbvralv 2533 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑦𝐵𝑧𝐶 𝜒)
4 cbvral3v.2 . . . 4 (𝑦 = 𝑣 → (𝜒𝜃))
5 cbvral3v.3 . . . 4 (𝑧 = 𝑢 → (𝜃𝜓))
64, 5cbvral2v 2541 . . 3 (∀𝑦𝐵𝑧𝐶 𝜒 ↔ ∀𝑣𝐵𝑢𝐶 𝜓)
76ralbii 2330 . 2 (∀𝑤𝐴𝑦𝐵𝑧𝐶 𝜒 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
83, 7bitri 173 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311 This theorem is referenced by: (None)
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