Theorem r19.23

Description: Restricted quantifier version of 19.23 . See r19.23v for a version requiring fewer axioms. (Contributed by NM, 22-Oct-2010) (Proof shortened by Mario Carneiro, 8-Oct-2016)

	Ref	Expression
Hypothesis	r19.23.1	⊢ Ⅎ 𝑥 𝜓
Assertion	r19.23	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.23.1	⊢ Ⅎ 𝑥 𝜓
2		r19.23t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) ) )
3	1 2	ax-mp	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )