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Theorem cbv3hv 2346
Description: Version of cbv3h 2438 with a dv condition on 𝑥, 𝑦, which does not require ax-13 2419. Was used in a proof of axc11n 2473 (but of independent interest). (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (Proof shortened by BJ, 30-Nov-2020.)
Hypotheses
Ref Expression
cbv3hv.nf1 (𝜑 → ∀𝑦𝜑)
cbv3hv.nf2 (𝜓 → ∀𝑥𝜓)
cbv3hv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbv3hv (∀𝑥𝜑 → ∀𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbv3hv
StepHypRef Expression
1 cbv3hv.nf1 . . 3 (𝜑 → ∀𝑦𝜑)
21nf5i 2189 . 2 𝑦𝜑
3 cbv3hv.nf2 . . 3 (𝜓 → ∀𝑥𝜓)
43nf5i 2189 . 2 𝑥𝜓
5 cbv3hv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
62, 4, 5cbv3v 2344 1 (∀𝑥𝜑 → ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-10 2184  ax-11 2200  ax-12 2213
This theorem depends on definitions:  df-bi 198  df-ex 1860  df-nf 1864
This theorem is referenced by: (None)
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