Theorem 19.30

Description: Theorem 19.30 of Margaris p. 90. (Contributed by NM, 12-Mar-1993) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Assertion	19.30	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) → ( ∀ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		exnal	⊢ ( ∃ 𝑥 ¬ 𝜑 ↔ ¬ ∀ 𝑥 𝜑 )
2		pm2.53	⊢ ( ( 𝜑 ∨ 𝜓 ) → ( ¬ 𝜑 → 𝜓 ) )
3	2	aleximi	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) → ( ∃ 𝑥 ¬ 𝜑 → ∃ 𝑥 𝜓 ) )
4	1 3	syl5bir	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) → ( ¬ ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
5	4	orrd	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) → ( ∀ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) )