Theorem r19.36v

Description: Restricted quantifier version of one direction of 19.36 . (The other direction holds iff A is nonempty, see r19.36zv .) (Contributed by NM, 22-Oct-2003)

	Ref	Expression
Assertion	r19.36v	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.35	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
2		id	⊢ ( 𝜓 → 𝜓 )
3	2	rexlimivw	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜓 )
4	3	imim2i	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )
5	1 4	sylbi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )