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

Theorem sepexi 5306
Description: Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5301. Inference associated with sepex 5305. (Contributed by NM, 21-Jun-1993.) Generalize conclusion, extract closed form, and avoid ax-9 2115. (Revised by Matthew House, 19-Sep-2025.)
Hypothesis
Ref Expression
sepexi.1 𝑦𝑥(𝜑𝑥𝑦)
Assertion
Ref Expression
sepexi 𝑧𝑥(𝑥𝑧𝜑)
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧   𝜑,𝑦   𝜑,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sepexi
StepHypRef Expression
1 sepexi.1 . 2 𝑦𝑥(𝜑𝑥𝑦)
2 sepex 5305 . 2 (∃𝑦𝑥(𝜑𝑥𝑦) → ∃𝑧𝑥(𝑥𝑧𝜑))
31, 2ax-mp 5 1 𝑧𝑥(𝑥𝑧𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1534  wex 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-sep 5301
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776
This theorem is referenced by:  axpow3  5373  vpwex  5382  axpr  5432  zfpair2  5438  axun2  7755  uniex2  7756
  Copyright terms: Public domain W3C validator