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Theorem cbvrex2 41596
Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3283. (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
cbvrex2 (∃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 cbvrex2
StepHypRef Expression
1 nfcv 2866 . . . 4 𝑧𝐵
2 cbvral2.1 . . . 4 𝑧𝜑
31, 2nfrex 3109 . . 3 𝑧𝑦𝐵 𝜑
4 nfcv 2866 . . . 4 𝑥𝐵
5 cbvral2.2 . . . 4 𝑥𝜒
64, 5nfrex 3109 . . 3 𝑥𝑦𝐵 𝜒
7 cbvral2.5 . . . 4 (𝑥 = 𝑧 → (𝜑𝜒))
87rexbidv 3154 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜒))
93, 6, 8cbvrex 3271 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑦𝐵 𝜒)
10 cbvral2.3 . . . 4 𝑤𝜒
11 cbvral2.4 . . . 4 𝑦𝜓
12 cbvral2.6 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
1310, 11, 12cbvrex 3271 . . 3 (∃𝑦𝐵 𝜒 ↔ ∃𝑤𝐵 𝜓)
1413rexbii 3143 . 2 (∃𝑧𝐴𝑦𝐵 𝜒 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
159, 14bitri 264 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1821  wrex 3015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020
This theorem is referenced by: (None)
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