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Theorem bj-cbv3v2 32702
 Description: Version of cbv3 2263 with two dv conditions, which does not require ax-11 2032 nor ax-13 2244. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbv3v2.nf 𝑥𝜓
bj-cbv3v2.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-cbv3v2 (∀𝑥𝜑 → ∀𝑦𝜓)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem bj-cbv3v2
StepHypRef Expression
1 nfv 1841 . 2 𝑦𝑥𝜑
2 bj-cbv3v2.nf . . 3 𝑥𝜓
3 bj-cbv3v2.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3spimv1 2113 . 2 (∀𝑥𝜑𝜓)
51, 4alrimi 2080 1 (∀𝑥𝜑 → ∀𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1479  Ⅎwnf 1706 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-12 2045 This theorem depends on definitions:  df-bi 197  df-ex 1703  df-nf 1708 This theorem is referenced by:  bj-cbv3hv2  32703
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