Theorem r19.36vf

Description: Restricted quantifier version of one direction of 19.36 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypothesis	r19.36vf.1	⊢ Ⅎ 𝑥 𝜓
Assertion	r19.36vf	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.36vf.1	⊢ Ⅎ 𝑥 𝜓
2		r19.35	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
3		idd	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜓 ) )
4	1 3	rexlimi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜓 )
5	4	imim2i	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )
6	2 5	sylbi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )