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Theorem cbvral2 43645
Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3414. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Hypotheses
Ref Expression
cbvral2.1 𝑧𝜑
cbvral2.2 𝑥𝜒
cbvral2.3 𝑤𝜒
cbvral2.4 𝑦𝜓
cbvral2.5 (𝑥 = 𝑧 → (𝜑𝜒))
cbvral2.6 (𝑦 = 𝑤 → (𝜒𝜓))
Assertion
Ref Expression
cbvral2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
Distinct variable groups:   𝑥,𝑧,𝐴   𝑥,𝑦,𝐵,𝑧   𝑦,𝑤,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)   𝜒(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑦,𝑤)

Proof of Theorem cbvral2
StepHypRef Expression
1 nfcv 2958 . . . 4 𝑧𝐵
2 cbvral2.1 . . . 4 𝑧𝜑
31, 2nfral 3193 . . 3 𝑧𝑦𝐵 𝜑
4 nfcv 2958 . . . 4 𝑥𝐵
5 cbvral2.2 . . . 4 𝑥𝜒
64, 5nfral 3193 . . 3 𝑥𝑦𝐵 𝜒
7 cbvral2.5 . . . 4 (𝑥 = 𝑧 → (𝜑𝜒))
87ralbidv 3165 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 𝜒))
93, 6, 8cbvralw 3390 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑦𝐵 𝜒)
10 cbvral2.3 . . . 4 𝑤𝜒
11 cbvral2.4 . . . 4 𝑦𝜓
12 cbvral2.6 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
1310, 11, 12cbvralw 3390 . . 3 (∀𝑦𝐵 𝜒 ↔ ∀𝑤𝐵 𝜓)
1413ralbii 3136 . 2 (∀𝑧𝐴𝑦𝐵 𝜒 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
159, 14bitri 278 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wnf 1785  wral 3109
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2382  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114
This theorem is referenced by: (None)
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