Theorem r19.36vf 43825

Description: Restricted quantifier version of one direction of 19.36 2224. (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 3109	. 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 3257	. . 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 1786   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787  df-ral 3063  df-rex 3072

This theorem is referenced by:  iinssf  43827