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Theorem r19.30 3336
 Description: Restricted quantifier version of 19.30 1876. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
Assertion
Ref Expression
r19.30 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))

Proof of Theorem r19.30
StepHypRef Expression
1 pm2.53 847 . . . 4 ((𝜓𝜑) → (¬ 𝜓𝜑))
21orcoms 868 . . 3 ((𝜑𝜓) → (¬ 𝜓𝜑))
32ralimi 3158 . 2 (∀𝑥𝐴 (𝜑𝜓) → ∀𝑥𝐴𝜓𝜑))
4 ralim 3160 . 2 (∀𝑥𝐴𝜓𝜑) → (∀𝑥𝐴 ¬ 𝜓 → ∀𝑥𝐴 𝜑))
5 ralnex 3234 . . . . . 6 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
65biimpri 230 . . . . 5 (¬ ∃𝑥𝐴 𝜓 → ∀𝑥𝐴 ¬ 𝜓)
76imim1i 63 . . . 4 ((∀𝑥𝐴 ¬ 𝜓 → ∀𝑥𝐴 𝜑) → (¬ ∃𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜑))
87orrd 859 . . 3 ((∀𝑥𝐴 ¬ 𝜓 → ∀𝑥𝐴 𝜑) → (∃𝑥𝐴 𝜓 ∨ ∀𝑥𝐴 𝜑))
98orcomd 867 . 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 843  ∀wral 3136  ∃wrex 3137 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1775  df-ral 3141  df-rex 3142 This theorem is referenced by:  disjunsn  30336  esumcvg  31338
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