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Theorem r19.30OLD 3119
Description: Obsolete version of 19.30 1879 as of 5-Nov-2024. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
r19.30OLD (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))

Proof of Theorem r19.30OLD
StepHypRef Expression
1 pm2.53 851 . . . 4 ((𝜓𝜑) → (¬ 𝜓𝜑))
21orcoms 872 . . 3 ((𝜑𝜓) → (¬ 𝜓𝜑))
32ralimi 3081 . 2 (∀𝑥𝐴 (𝜑𝜓) → ∀𝑥𝐴𝜓𝜑))
4 ralim 3084 . 2 (∀𝑥𝐴𝜓𝜑) → (∀𝑥𝐴 ¬ 𝜓 → ∀𝑥𝐴 𝜑))
5 ralnex 3070 . . . . . 6 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
65biimpri 228 . . . . 5 (¬ ∃𝑥𝐴 𝜓 → ∀𝑥𝐴 ¬ 𝜓)
76imim1i 63 . . . 4 ((∀𝑥𝐴 ¬ 𝜓 → ∀𝑥𝐴 𝜑) → (¬ ∃𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜑))
87orrd 863 . . 3 ((∀𝑥𝐴 ¬ 𝜓 → ∀𝑥𝐴 𝜑) → (∃𝑥𝐴 𝜓 ∨ ∀𝑥𝐴 𝜑))
98orcomd 871 . 2 ((∀𝑥𝐴 ¬ 𝜓 → ∀𝑥𝐴 𝜑) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
103, 4, 93syl 18 1 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 847  wral 3059  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-ral 3060  df-rex 3069
This theorem is referenced by: (None)
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