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Theorem 19.40b 1804
Description: The antecedent provides a condition implying the converse of 19.40 1785. This is to 19.40 1785 what 19.33b 1802 is to 19.33 1801. (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 pm3.21 463 . . . . 5 (𝜓 → (𝜑 → (𝜑𝜓)))
21aleximi 1749 . . . 4 (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
3 pm3.2 462 . . . . 5 (𝜑 → (𝜓 → (𝜑𝜓)))
43aleximi 1749 . . . 4 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
52, 4jaoa 531 . . 3 ((∀𝑥𝜓 ∨ ∀𝑥𝜑) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓)))
65orcoms 403 . 2 ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓)))
7 19.40 1785 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
86, 7impbid1 214 1 ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  wa 383  wal 1473  wex 1695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696
This theorem is referenced by: (None)
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