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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 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.37v
StepHypRef Expression
1 nfv 1883 . 2 𝑥𝜑
21r19.37 3115 1 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 Colors of variables: wff setvar class 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
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