Theorem r19.37v 3344

Description: Restricted quantifier version of one direction of 19.37v 1994. (The other direction holds iff 𝐴 is nonempty, see r19.37zv 4447.) (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 22	. . 3 ⊢ (𝜑 → 𝜑)
2	1	ralrimivw 3183	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
3		r19.35 3341	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
4	3	biimpi 218	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
5	2, 4	syl5 34	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4  ∀wral 3138  ∃wrex 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-ral 3143  df-rex 3144

This theorem is referenced by:  ssiun  4963  isucn2  22882