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(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))

Proof of Theorem 19.30

Step	Hyp	Ref	Expression
1		exnal 1829	. . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
2		pm2.53 848	. . . 4 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
3	2	aleximi 1834	. . 3 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∃𝑥 ¬ 𝜑 → ∃𝑥𝜓))
4	1, 3	syl5bir 242	. 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (¬ ∀𝑥𝜑 → ∃𝑥𝜓))
5	4	orrd 860	1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))

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