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Theorem 19.43OLD 1800
 Description: Obsolete proof of 19.43 1799. Do not delete as it is referenced on the mmrecent.html page and in conventions-label 26651. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
19.43OLD (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Proof of Theorem 19.43OLD
StepHypRef Expression
1 ioran 510 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
21albii 1737 . . . 4 (∀𝑥 ¬ (𝜑𝜓) ↔ ∀𝑥𝜑 ∧ ¬ 𝜓))
3 19.26 1786 . . . 4 (∀𝑥𝜑 ∧ ¬ 𝜓) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ¬ 𝜓))
4 alnex 1697 . . . . 5 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5 alnex 1697 . . . . 5 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
64, 5anbi12i 729 . . . 4 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ¬ 𝜓) ↔ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
72, 3, 63bitri 285 . . 3 (∀𝑥 ¬ (𝜑𝜓) ↔ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
87notbii 309 . 2 (¬ ∀𝑥 ¬ (𝜑𝜓) ↔ ¬ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
9 df-ex 1696 . 2 (∃𝑥(𝜑𝜓) ↔ ¬ ∀𝑥 ¬ (𝜑𝜓))
10 oran 516 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ¬ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
118, 9, 103bitr4i 291 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ 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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