Theorem r19.37v 3189

Description: Restricted quantifier version of one direction of 19.37v 2030. (The other direction holds iff 𝐴 is nonempty, see r19.37zv 4463.) (Contributed by NM, 2-Apr-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Jun-2023.)

Assertion

Ref	Expression
r19.37v	⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.37v

Step	Hyp	Ref	Expression
1		id 23	. . 3 ⊢ (𝜑 → 𝜑)
2	1	ralrimivw 3159	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
3		r19.35 3121	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
4	3	biimpi 219	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
5	2, 4	syl5 35	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wral 3077  ∃wrex 3087

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

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3078  df-rex 3088

This theorem is used by:  ssiun  5005  isucn2  24597