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Theorem bj-cbv3ta 32349
Description: Closed form of cbv3 2264. (Contributed by BJ, 2-May-2019.)
Assertion
Ref Expression
bj-cbv3ta (∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦(∃𝑥𝜓𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Proof of Theorem bj-cbv3ta
StepHypRef Expression
1 bj-spimt2 32348 . . . . . 6 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ((∃𝑥𝜓𝜓) → (∀𝑥𝜑𝜓)))
21imp 445 . . . . 5 ((∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) ∧ (∃𝑥𝜓𝜓)) → (∀𝑥𝜑𝜓))
32alanimi 1741 . . . 4 ((∀𝑦𝑥(𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∀𝑦(∃𝑥𝜓𝜓)) → ∀𝑦(∀𝑥𝜑𝜓))
4 bj-hbalt 32310 . . . 4 (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝑥𝜑))
5 sylgt 1746 . . . 4 (∀𝑦(∀𝑥𝜑𝜓) → ((∀𝑥𝜑 → ∀𝑦𝑥𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
63, 4, 5syl2im 40 . . 3 ((∀𝑦𝑥(𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∀𝑦(∃𝑥𝜓𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
76expimpd 628 . 2 (∀𝑦𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦(∃𝑥𝜓𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
87alcoms 2032 1 (∀𝑥𝑦(𝑥 = 𝑦 → (𝜑𝜓)) → ((∀𝑦(∃𝑥𝜓𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  bj-cbv3tb  32350
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