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

Theorem hbeu1 1958
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 1951 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 hba1 1478 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑥(𝜑𝑥 = 𝑦))
32hbex 1572 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑦𝑥(𝜑𝑥 = 𝑦))
41, 3hbxfrbi 1406 1 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wex 1426  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-eu 1951
This theorem is referenced by:  hbmo1  1986  eupicka  2028  exists2  2045
  Copyright terms: Public domain W3C validator