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Theorem bj-cbv2hv 34229
Description: Version of cbv2h 2418 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbv2hv.1 (𝜑 → (𝜓 → ∀𝑦𝜓))
bj-cbv2hv.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
bj-cbv2hv.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
bj-cbv2hv (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem bj-cbv2hv
StepHypRef Expression
1 bj-cbv2hv.1 . . 3 (𝜑 → (𝜓 → ∀𝑦𝜓))
2 bj-cbv2hv.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
3 bj-cbv2hv.3 . . . 4 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
4 biimp 218 . . . 4 ((𝜓𝜒) → (𝜓𝜒))
53, 4syl6 35 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
61, 2, 5bj-cbv1hv 34228 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
7 equcomi 2024 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
8 biimpr 223 . . . . 5 ((𝜓𝜒) → (𝜒𝜓))
97, 3, 8syl56 36 . . . 4 (𝜑 → (𝑦 = 𝑥 → (𝜒𝜓)))
102, 1, 9bj-cbv1hv 34228 . . 3 (∀𝑦𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
1110alcoms 2160 . 2 (∀𝑥𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
126, 11impbid 215 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by:  bj-cbv2v  34230
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