ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  hbeu1 GIF version

Theorem hbeu1 2016
Description: Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
Assertion
Ref Expression
hbeu1 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)

Proof of Theorem hbeu1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-eu 2009 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 hba1 1520 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑥(𝜑𝑥 = 𝑦))
32hbex 1616 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑦𝑥(𝜑𝑥 = 𝑦))
41, 3hbxfrbi 1452 1 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1333  wex 1472  ∃!weu 2006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-eu 2009
This theorem is referenced by:  hbmo1  2044  eupicka  2086  exists2  2103
  Copyright terms: Public domain W3C validator