 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.37v Structured version   Visualization version   GIF version

Theorem r19.37v 3234
 Description: Restricted quantifier version of one direction of 19.37v 2077. (The other direction holds iff 𝐴 is nonempty, see r19.37zv 4206.) (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 1994 . 2 𝑥𝜑
21r19.37 3233 1 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wrex 3061 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-12 2202 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-nf 1857  df-ral 3065  df-rex 3066 This theorem is referenced by:  ssiun  4694  isucn2  22302
 Copyright terms: Public domain W3C validator