Theorem r19.21

Description: Restricted quantifier version of 19.21 . (Contributed by Scott Fenton, 30-Mar-2011)

	Ref	Expression
Hypothesis	r19.21.1	⊢ Ⅎ 𝑥 𝜑
Assertion	r19.21	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.21.1	⊢ Ⅎ 𝑥 𝜑
2		r19.21t	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) )
3	1 2	ax-mp	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) )