Theorem r19.30

Description: Restricted quantifier version of 19.30 . (Contributed by Scott Fenton, 25-Feb-2011) (Proof shortened by Wolf Lammen, 5-Nov-2024)

	Ref	Expression
Assertion	r19.30	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		pm2.53	⊢ ( ( 𝜑 ∨ 𝜓 ) → ( ¬ 𝜑 → 𝜓 ) )
2	1	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ∀ 𝑥 ∈ 𝐴 ( ¬ 𝜑 → 𝜓 ) )
3		rexnal	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 𝜑 )
4	3	biimpri	⊢ ( ¬ ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 )
5		rexim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( ¬ 𝜑 → 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
6	2 4 5	syl2im	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ( ¬ ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
7	6	orrd	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )