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Theorem cbvrex2 44012
 Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3375. (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,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 cbvrex2
StepHypRef Expression
1 nfcv 2917 . . . 4 𝑧𝐵
2 cbvral2.1 . . . 4 𝑧𝜑
31, 2nfrex 3231 . . 3 𝑧𝑦𝐵 𝜑
4 nfcv 2917 . . . 4 𝑥𝐵
5 cbvral2.2 . . . 4 𝑥𝜒
64, 5nfrex 3231 . . 3 𝑥𝑦𝐵 𝜒
7 cbvral2.5 . . . 4 (𝑥 = 𝑧 → (𝜑𝜒))
87rexbidv 3219 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜒))
93, 6, 8cbvrexw 3351 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑦𝐵 𝜒)
10 cbvral2.3 . . . 4 𝑤𝜒
11 cbvral2.4 . . . 4 𝑦𝜓
12 cbvral2.6 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
1310, 11, 12cbvrexw 3351 . . 3 (∃𝑦𝐵 𝜒 ↔ ∃𝑤𝐵 𝜓)
1413rexbii 3173 . 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  Ⅎwnf 1786  ∃wrex 3069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ral 3073  df-rex 3074 This theorem is referenced by: (None)
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