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

Theorem euf 1905
 Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)
Hypothesis
Ref Expression
euf.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
euf (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem euf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-eu 1903 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 euf.1 . . . . 5 (𝜑 → ∀𝑦𝜑)
3 ax-17 1419 . . . . 5 (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)
42, 3hbbi 1440 . . . 4 ((𝜑𝑥 = 𝑧) → ∀𝑦(𝜑𝑥 = 𝑧))
54hbal 1366 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑦𝑥(𝜑𝑥 = 𝑧))
6 ax-17 1419 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧𝑥(𝜑𝑥 = 𝑦))
7 equequ2 1599 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87bibi2d 221 . . . 4 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1705 . . 3 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 6, 9cbvexh 1638 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
111, 10bitri 173 1 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381  ∃!weu 1900 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-eu 1903 This theorem is referenced by:  eu1  1925  eumo0  1931
 Copyright terms: Public domain W3C validator