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

Theorem exsimpr 1870
 Description: Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
exsimpr (∃𝑥(𝜑𝜓) → ∃𝑥𝜓)

Proof of Theorem exsimpr
StepHypRef Expression
1 simpr 488 . 2 ((𝜑𝜓) → 𝜓)
21eximi 1836 1 (∃𝑥(𝜑𝜓) → ∃𝑥𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  19.40  1887  spsbeALT  2565  rexex  3203  ceqsexv2d  3490  imassrn  5907  fv3  6663  finacn  9461  dfac4  9533  kmlem2  9562  ac6c5  9893  ac6s3  9898  ac6s5  9902  bj-finsumval0  34697  mptsnunlem  34752  topdifinffinlem  34761  heiborlem3  35248  ac6s3f  35606  moantr  35773
 Copyright terms: Public domain W3C validator