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Theorem 19.40bOLD 1805
 Description: Obsolete proof of 19.40b 1804 as of 13-Nov-2020. (Contributed by BJ, 6-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
19.40bOLD ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓)))

Proof of Theorem 19.40bOLD
StepHypRef Expression
1 19.29 1789 . . . . 5 ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
21ex 449 . . . 4 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
32adantld 482 . . 3 (∀𝑥𝜑 → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓)))
4 19.29r 1790 . . . . 5 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
54expcom 450 . . . 4 (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
65adantrd 483 . . 3 (∀𝑥𝜓 → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓)))
73, 6jaoi 393 . 2 ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓)))
8 19.40 1785 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
97, 8impbid1 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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