Theorem r19.37v 3116

Description: Restricted quantifier version of one direction of 19.37v 1966. (The other direction holds iff 𝐴 is nonempty, see r19.37zv 4100.) (Contributed by NM, 2-Apr-2004.)

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		nfv 1883	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.37 3115	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4  ∃wrex 2942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750  df-ral 2946  df-rex 2947

This theorem is referenced by:  ssiun  4594  isucn2  22130