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Theorem cbv3hv 2172
Description: Version of cbv3h 2264 with a dv condition on 𝑥, 𝑦, which does not require ax-13 2244. Was used in a proof of axc11n 2305 (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 2022 . 2 𝑦𝜑
3 cbv3hv.nf2 . . 3 (𝜓 → ∀𝑥𝜓)
43nf5i 2022 . 2 𝑥𝜓
5 cbv3hv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
62, 4, 5cbv3v 2170 1 (∀𝑥𝜑 → ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-ex 1703  df-nf 1708
This theorem is referenced by: (None)
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