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Theorem bj-cbv3tb 31898
 Description: Closed form of cbv3 2253. (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 2059 . . . 4 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
21biimpd 218 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
32alimi 1730 . 2 (∀𝑦𝑥𝜓 → ∀𝑦(∃𝑥𝜓𝜓))
4 nf5r 2052 . . 3 (Ⅎ𝑦𝜑 → (𝜑 → ∀𝑦𝜑))
54alimi 1730 . 2 (∀𝑥𝑦𝜑 → ∀𝑥(𝜑 → ∀𝑦𝜑))
6 bj-cbv3ta 31897 . 2 (∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦(∃𝑥𝜓𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
73, 5, 6syl2ani 686 1 (∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦𝑥𝜓 ∧ ∀𝑥𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695  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-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by: (None)
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