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Theorem bj-cbv1hv 34111
Description: Version of cbv1h 2418 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbv1hv.1 (𝜑 → (𝜓 → ∀𝑦𝜓))
bj-cbv1hv.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
bj-cbv1hv.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
bj-cbv1hv (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem bj-cbv1hv
StepHypRef Expression
1 nfa1 2148 . 2 𝑥𝑥𝑦𝜑
2 nfa2 2168 . 2 𝑦𝑥𝑦𝜑
3 2sp 2177 . . . 4 (∀𝑥𝑦𝜑𝜑)
4 bj-cbv1hv.1 . . . 4 (𝜑 → (𝜓 → ∀𝑦𝜓))
53, 4syl 17 . . 3 (∀𝑥𝑦𝜑 → (𝜓 → ∀𝑦𝜓))
62, 5nf5d 2285 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜓)
7 bj-cbv1hv.2 . . . 4 (𝜑 → (𝜒 → ∀𝑥𝜒))
83, 7syl 17 . . 3 (∀𝑥𝑦𝜑 → (𝜒 → ∀𝑥𝜒))
91, 8nf5d 2285 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑥𝜒)
10 bj-cbv1hv.3 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
113, 10syl 17 . 2 (∀𝑥𝑦𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
121, 2, 6, 9, 11cbv1v 2349 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528
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 209  df-or 844  df-ex 1774  df-nf 1778
This theorem is referenced by:  bj-cbv2hv  34112
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