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Theorem bj-cbv2v 33097
Description: Version of cbv2 2372 with a disjoint variable condition, which does not require ax-13 2349. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbv2v.1 𝑥𝜑
bj-cbv2v.2 𝑦𝜑
bj-cbv2v.3 (𝜑 → Ⅎ𝑦𝜓)
bj-cbv2v.4 (𝜑 → Ⅎ𝑥𝜒)
bj-cbv2v.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
bj-cbv2v (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem bj-cbv2v
StepHypRef Expression
1 bj-cbv2v.2 . . . 4 𝑦𝜑
21nf5ri 2227 . . 3 (𝜑 → ∀𝑦𝜑)
3 bj-cbv2v.1 . . . . 5 𝑥𝜑
43nfal 2323 . . . 4 𝑥𝑦𝜑
54nf5ri 2227 . . 3 (∀𝑦𝜑 → ∀𝑥𝑦𝜑)
62, 5syl 17 . 2 (𝜑 → ∀𝑥𝑦𝜑)
7 bj-cbv2v.3 . . . 4 (𝜑 → Ⅎ𝑦𝜓)
87nf5rd 2228 . . 3 (𝜑 → (𝜓 → ∀𝑦𝜓))
9 bj-cbv2v.4 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
109nf5rd 2228 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
11 bj-cbv2v.5 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
128, 10, 11bj-cbv2hv 33096 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
136, 12syl 17 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650  wnf 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879
This theorem is referenced by:  bj-cbvaldv  33100
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