Theorem r19.36vf 42638

Description: Restricted quantifier version of one direction of 19.36 2226. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
r19.36vf.1	⊢ Ⅎ𝑥𝜓

Assertion

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

Proof of Theorem r19.36vf

Step	Hyp	Ref	Expression
1		r19.35 3270	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
2		r19.36vf.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3		idd 24	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜓 → 𝜓))
4	2, 3	rexlimi 3245	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝜓)
5	4	imim2i 16	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))
6	1, 5	sylbi 216	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))

Syntax hints:   → wi 4  Ⅎwnf 1789   ∈ wcel 2109  ∀wral 3065  ∃wrex 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-nf 1790  df-ral 3070  df-rex 3071

This theorem is referenced by:  iinssf  42640