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

Theorem exlimih 2133
 Description: Inference associated with 19.23 2067. See exlimiv 1845 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
Hypotheses
Ref Expression
exlimih.1 (𝜓 → ∀𝑥𝜓)
exlimih.2 (𝜑𝜓)
Assertion
Ref Expression
exlimih (∃𝑥𝜑𝜓)

Proof of Theorem exlimih
StepHypRef Expression
1 exlimih.1 . . 3 (𝜓 → ∀𝑥𝜓)
21nf5i 2011 . 2 𝑥𝜓
3 exlimih.2 . 2 (𝜑𝜓)
42, 3exlimi 2073 1 (∃𝑥𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator