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Theorem 19.40b 1812
 Description: The antecedent provides a condition implying the converse of 19.40 1794. This is to 19.40 1794 what 19.33b 1810 is to 19.33 1809. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.)
Assertion
Ref Expression
19.40b ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓)))

Proof of Theorem 19.40b
StepHypRef Expression
1 pm3.21 464 . . . . 5 (𝜓 → (𝜑 → (𝜑𝜓)))
21aleximi 1756 . . . 4 (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
3 pm3.2 463 . . . . 5 (𝜑 → (𝜓 → (𝜑𝜓)))
43aleximi 1756 . . . 4 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
52, 4jaoa 532 . . 3 ((∀𝑥𝜓 ∨ ∀𝑥𝜑) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓)))
65orcoms 404 . 2 ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓)))
7 19.40 1794 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
86, 7impbid1 215 1 ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ 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 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702 This theorem is referenced by: (None)
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