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Theorem cbv3v 2347
Description: Rule used to change bound variables, using implicit substitution. Version of cbv3 2407 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
Hypotheses
Ref Expression
cbv3v.nf1 𝑦𝜑
cbv3v.nf2 𝑥𝜓
cbv3v.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbv3v (∀𝑥𝜑 → ∀𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbv3v
StepHypRef Expression
1 cbv3v.nf1 . . . 4 𝑦𝜑
21nf5ri 2194 . . 3 (𝜑 → ∀𝑦𝜑)
32hbal 2172 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
4 cbv3v.nf2 . . 3 𝑥𝜓
5 cbv3v.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
64, 5spimfv 2240 . 2 (∀𝑥𝜑𝜓)
73, 6alrimih 1825 1 (∀𝑥𝜑 → ∀𝑦𝜓)
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 1911  ax-6 1970  ax-7 2015  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786
This theorem is referenced by:  cbv1v  2348  cbv3hv  2352  cbvalv1  2353
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