Theorem r19.37

Description: Restricted quantifier version of one direction of 19.37 . (The other direction does not hold when A is empty.) (Contributed by FL, 13-May-2012) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypothesis	r19.37.1	⊢ Ⅎ 𝑥 𝜑
Assertion	r19.37	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.37.1	⊢ Ⅎ 𝑥 𝜑
2		r19.35	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
3		ax-1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
4	1 3	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 )
5	4	imim1i	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) → ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
6	2 5	sylbi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )