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Theorem bj-cbv3tb 32686
Description: Closed form of cbv3 2263. (Contributed by BJ, 2-May-2019.)
Assertion
Ref Expression
bj-cbv3tb (∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦𝑥𝜓 ∧ ∀𝑥𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Proof of Theorem bj-cbv3tb
StepHypRef Expression
1 19.9t 2069 . . . 4 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
21biimpd 219 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
32alimi 1737 . 2 (∀𝑦𝑥𝜓 → ∀𝑦(∃𝑥𝜓𝜓))
4 nf5r 2062 . . 3 (Ⅎ𝑦𝜑 → (𝜑 → ∀𝑦𝜑))
54alimi 1737 . 2 (∀𝑥𝑦𝜑 → ∀𝑥(𝜑 → ∀𝑦𝜑))
6 bj-cbv3ta 32685 . 2 (∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦(∃𝑥𝜓𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
73, 5, 6syl2ani 687 1 (∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦𝑥𝜓 ∧ ∀𝑥𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1479  wex 1702  wnf 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-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-nf 1708
This theorem is referenced by: (None)
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