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Theorem bj-cbv2v 34023
 Description: Version of cbv2 2419 with a disjoint variable condition, which does not require ax-13 2385. (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 2187 . . 3 (𝜑 → ∀𝑦𝜑)
3 bj-cbv2v.1 . . . . 5 𝑥𝜑
43nfal 2336 . . . 4 𝑥𝑦𝜑
54nf5ri 2187 . . 3 (∀𝑦𝜑 → ∀𝑥𝑦𝜑)
62, 5syl 17 . 2 (𝜑 → ∀𝑥𝑦𝜑)
7 bj-cbv2v.3 . . . 4 (𝜑 → Ⅎ𝑦𝜓)
87nf5rd 2189 . . 3 (𝜑 → (𝜓 → ∀𝑦𝜓))
9 bj-cbv2v.4 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
109nf5rd 2189 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
11 bj-cbv2v.5 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
128, 10, 11bj-cbv2hv 34022 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
136, 12syl 17 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1528  Ⅎwnf 1777 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-nf 1778 This theorem is referenced by:  bj-cbvaldv  34024
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