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Theorem 19.30 1849
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 1794 . . 3 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
2 pm2.53 387 . . . 4 ((𝜑𝜓) → (¬ 𝜑𝜓))
32aleximi 1799 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥 ¬ 𝜑 → ∃𝑥𝜓))
41, 3syl5bir 233 . 2 (∀𝑥(𝜑𝜓) → (¬ ∀𝑥𝜑 → ∃𝑥𝜓))
54orrd 392 1 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777
This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1745
This theorem is referenced by:  19.33b  1853
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