Theorem r19.36vf 45887

Description: Restricted quantifier version of one direction of 19.36 2269. (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 3125	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
2		r19.36vf.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3		idd 25	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜓 → 𝜓))
4	2, 3	rexlimi 3267	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝜓)
5	4	imim2i 17	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))
6	1, 5	sylbi 220	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1816   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-ral 3082  df-rex 3092

This theorem is used by:  iinssf  45889