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Theorem bj-cbv1v 32854
Description: Version of cbv1 2303 with a dv condition, which does not require ax-13 2282. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbv1v.1 𝑥𝜑
bj-cbv1v.2 𝑦𝜑
bj-cbv1v.3 (𝜑 → Ⅎ𝑦𝜓)
bj-cbv1v.4 (𝜑 → Ⅎ𝑥𝜒)
bj-cbv1v.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
bj-cbv1v (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem bj-cbv1v
StepHypRef Expression
1 bj-cbv1v.2 . . . . 5 𝑦𝜑
2 bj-cbv1v.3 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
31, 2nfim1 2105 . . . 4 𝑦(𝜑𝜓)
4 bj-cbv1v.1 . . . . 5 𝑥𝜑
5 bj-cbv1v.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
64, 5nfim1 2105 . . . 4 𝑥(𝜑𝜒)
7 bj-cbv1v.5 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
87com12 32 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
98a2d 29 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) → (𝜑𝜒)))
103, 6, 9cbv3v 2208 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑦(𝜑𝜒))
11419.21 2113 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
12119.21 2113 . . 3 (∀𝑦(𝜑𝜒) ↔ (𝜑 → ∀𝑦𝜒))
1310, 11, 123imtr3i 280 . 2 ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑦𝜒))
1413pm2.86i 109 1 (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  bj-cbv1hv  32855
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