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Theorem 19.43OLD 1808
Description: Obsolete proof of 19.43 1807. Do not delete as it is referenced on the mmrecent.html page and in conventions-label 27113. (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 511 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
21albii 1744 . . . 4 (∀𝑥 ¬ (𝜑𝜓) ↔ ∀𝑥𝜑 ∧ ¬ 𝜓))
3 19.26 1795 . . . 4 (∀𝑥𝜑 ∧ ¬ 𝜓) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ¬ 𝜓))
4 alnex 1703 . . . . 5 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5 alnex 1703 . . . . 5 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
64, 5anbi12i 732 . . . 4 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ¬ 𝜓) ↔ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
72, 3, 63bitri 286 . . 3 (∀𝑥 ¬ (𝜑𝜓) ↔ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
87notbii 310 . 2 (¬ ∀𝑥 ¬ (𝜑𝜓) ↔ ¬ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
9 df-ex 1702 . 2 (∃𝑥(𝜑𝜓) ↔ ¬ ∀𝑥 ¬ (𝜑𝜓))
10 oran 517 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ¬ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
118, 9, 103bitr4i 292 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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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