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Theorem bj-cbv3tb 31701
Description: Closed form of cbv3 2248. (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 2057 . . . 4 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
21biimpd 217 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
32alimi 1729 . 2 (∀𝑦𝑥𝜓 → ∀𝑦(∃𝑥𝜓𝜓))
4 nf5r 2050 . . 3 (Ⅎ𝑦𝜑 → (𝜑 → ∀𝑦𝜑))
54alimi 1729 . 2 (∀𝑥𝑦𝜑 → ∀𝑥(𝜑 → ∀𝑦𝜑))
6 bj-cbv3ta 31700 . 2 (∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦(∃𝑥𝜓𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
73, 5, 6syl2ani 685 1 (∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦𝑥𝜓 ∧ ∀𝑥𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1472  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-11 2020  ax-12 2032  ax-13 2229
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-nf 1700
This theorem is referenced by: (None)
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