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Theorem 19.43OLD 1884
Description: Obsolete proof of 19.43 1883. Do not delete as it is referenced on the mmrecent.html 1883 page and in conventions-labels 28186. (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 981 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
21albii 1821 . . . 4 (∀𝑥 ¬ (𝜑𝜓) ↔ ∀𝑥𝜑 ∧ ¬ 𝜓))
3 19.26 1871 . . . 4 (∀𝑥𝜑 ∧ ¬ 𝜓) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ¬ 𝜓))
4 alnex 1783 . . . . 5 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5 alnex 1783 . . . . 5 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
64, 5anbi12i 629 . . . 4 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ¬ 𝜓) ↔ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
72, 3, 63bitri 300 . . 3 (∀𝑥 ¬ (𝜑𝜓) ↔ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
87notbii 323 . 2 (¬ ∀𝑥 ¬ (𝜑𝜓) ↔ ¬ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
9 df-ex 1782 . 2 (∃𝑥(𝜑𝜓) ↔ ¬ ∀𝑥 ¬ (𝜑𝜓))
10 oran 987 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ¬ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
118, 9, 103bitr4i 306 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wo 844  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782
This theorem is referenced by: (None)
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