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Theorem cbvral2 39821
 Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3155. (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,𝐴   𝑧,𝐴   𝑥,𝑦,𝐵   𝑦,𝑧,𝐵   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 2751 . . . 4 𝑧𝐵
2 cbvral2.1 . . . 4 𝑧𝜑
31, 2nfral 2929 . . 3 𝑧𝑦𝐵 𝜑
4 nfcv 2751 . . . 4 𝑥𝐵
5 cbvral2.2 . . . 4 𝑥𝜒
64, 5nfral 2929 . . 3 𝑥𝑦𝐵 𝜒
7 cbvral2.5 . . . 4 (𝑥 = 𝑧 → (𝜑𝜒))
87ralbidv 2969 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 𝜒))
93, 6, 8cbvral 3143 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑦𝐵 𝜒)
10 cbvral2.3 . . . 4 𝑤𝜒
11 cbvral2.4 . . . 4 𝑦𝜓
12 cbvral2.6 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
1310, 11, 12cbvral 3143 . . 3 (∀𝑦𝐵 𝜒 ↔ ∀𝑤𝐵 𝜓)
1413ralbii 2963 . 2 (∀𝑧𝐴𝑦𝐵 𝜒 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
159, 14bitri 263 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  Ⅎwnf 1699  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901 This theorem is referenced by: (None)
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