Theorem r19.30

Description: Restricted quantifier version of 19.30 . (Contributed by Scott Fenton, 25-Feb-2011) (Proof shortened by Wolf Lammen, 18-Jun-2023)

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

Proof

Step	Hyp	Ref	Expression
1		pm2.53	⊢ ( ( 𝜓 ∨ 𝜑 ) → ( ¬ 𝜓 → 𝜑 ) )
2	1	orcoms	⊢ ( ( 𝜑 ∨ 𝜓 ) → ( ¬ 𝜓 → 𝜑 ) )
3	2	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ∀ 𝑥 ∈ 𝐴 ( ¬ 𝜓 → 𝜑 ) )
4		ralim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( ¬ 𝜓 → 𝜑 ) → ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )
5		ralnex	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜓 )
6	5	biimpri	⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 )
7	6	imim1i	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜑 ) → ( ¬ ∃ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )
8	7	orrd	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜑 ) → ( ∃ 𝑥 ∈ 𝐴 𝜓 ∨ ∀ 𝑥 ∈ 𝐴 𝜑 ) )
9	8	orcomd	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜑 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )
10	3 4 9	3syl	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )