Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  euor2 GIF version

Theorem euor2 2058
 Description: Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
euor2 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))

Proof of Theorem euor2
StepHypRef Expression
1 hbe1 1472 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
21hbn 1633 . 2 (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)
3 19.8a 1570 . . . 4 (𝜑 → ∃𝑥𝜑)
43con3i 622 . . 3 (¬ ∃𝑥𝜑 → ¬ 𝜑)
5 orel1 715 . . . 4 𝜑 → ((𝜑𝜓) → 𝜓))
6 olc 701 . . . 4 (𝜓 → (𝜑𝜓))
75, 6impbid1 141 . . 3 𝜑 → ((𝜑𝜓) ↔ 𝜓))
84, 7syl 14 . 2 (¬ ∃𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
92, 8eubidh 2006 1 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 698  ∃wex 1469  ∃!weu 2000 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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-eu 2003 This theorem is referenced by:  reuun2  3365
 Copyright terms: Public domain W3C validator