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Theorem cbv1v 2358
Description: Rule used to change bound variables, using implicit substitution. Version of cbv1 2424 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.)
Hypotheses
Ref Expression
cbv1v.1 𝑥𝜑
cbv1v.2 𝑦𝜑
cbv1v.3 (𝜑 → Ⅎ𝑦𝜓)
cbv1v.4 (𝜑 → Ⅎ𝑥𝜒)
cbv1v.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv1v (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem cbv1v
StepHypRef Expression
1 cbv1v.2 . . . . 5 𝑦𝜑
2 cbv1v.3 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
31, 2nfim1 2201 . . . 4 𝑦(𝜑𝜓)
4 cbv1v.1 . . . . 5 𝑥𝜑
5 cbv1v.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
64, 5nfim1 2201 . . . 4 𝑥(𝜑𝜒)
7 cbv1v.5 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
87com12 32 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
98a2d 29 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) → (𝜑𝜒)))
103, 6, 9cbv3v 2357 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑦(𝜑𝜒))
11419.21 2209 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
12119.21 2209 . . 3 (∀𝑦(𝜑𝜒) ↔ (𝜑 → ∀𝑦𝜒))
1310, 11, 123imtr3i 294 . 2 ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑦𝜒))
1413pm2.86i 110 1 (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wnf 1785
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 1912  ax-6 1971  ax-7 2016  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by:  cbv2w  2359  vtoclgft  3539  bj-cbv1hv  34177
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