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Theorem cbv3v 2158
 Description: Version of cbv3 2253 with a dv condition, which does not require ax-13 2234. (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 2139 . 2 𝑦𝑥𝜑
3 cbv3v.nf2 . . 3 𝑥𝜓
4 cbv3v.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4spimv1 2101 . 2 (∀𝑥𝜑𝜓)
62, 5alrimi 2069 1 (∀𝑥𝜑 → ∀𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  cbv3hv  2160  cbvalv1  2163  bj-cbv1v  31916
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