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Theorem cbv3v 2169
Description: Version of cbv3 2264 with a dv condition, which does not require ax-13 2245. (Contributed 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 . . 3 𝑦𝜑
21nfal 2150 . 2 𝑦𝑥𝜑
3 cbv3v.nf2 . . 3 𝑥𝜓
4 cbv3v.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4spimv1 2112 . 2 (∀𝑥𝜑𝜓)
62, 5alrimi 2080 1 (∀𝑥𝜑 → ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707
This theorem is referenced by:  cbv3hv  2171  cbvalv1  2174  bj-cbv1v  32424
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