Theorem r19.32v

Description: Restricted quantifier version of 19.32v . (Contributed by NM, 25-Nov-2003)

	Ref	Expression
Assertion	r19.32v	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∀ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.21v	⊢ ( ∀ 𝑥 ∈ 𝐴 ( ¬ 𝜑 → 𝜓 ) ↔ ( ¬ 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) )
2		df-or	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 → 𝜓 ) )
3	2	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ( ¬ 𝜑 → 𝜓 ) )
4		df-or	⊢ ( ( 𝜑 ∨ ∀ 𝑥 ∈ 𝐴 𝜓 ) ↔ ( ¬ 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) )
5	1 3 4	3bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∀ 𝑥 ∈ 𝐴 𝜓 ) )