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Theorem 19.30 1884
Description: Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
19.30 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))

Proof of Theorem 19.30
StepHypRef Expression
1 exnal 1829 . . 3 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
2 pm2.53 848 . . . 4 ((𝜑𝜓) → (¬ 𝜑𝜓))
32aleximi 1834 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥 ¬ 𝜑 → ∃𝑥𝜓))
41, 3syl5bir 242 . 2 (∀𝑥(𝜑𝜓) → (¬ ∀𝑥𝜑 → ∃𝑥𝜓))
54orrd 860 1 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 844  wal 1537  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783
This theorem is referenced by:  19.33b  1888
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