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

Theorem euor2 2084
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 1495 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
21hbn 1654 . 2 (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)
3 19.8a 1590 . . . 4 (𝜑 → ∃𝑥𝜑)
43con3i 632 . . 3 (¬ ∃𝑥𝜑 → ¬ 𝜑)
5 orel1 725 . . . 4 𝜑 → ((𝜑𝜓) → 𝜓))
6 olc 711 . . . 4 (𝜓 → (𝜑𝜓))
75, 6impbid1 142 . . 3 𝜑 → ((𝜑𝜓) ↔ 𝜓))
84, 7syl 14 . 2 (¬ ∃𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
92, 8eubidh 2032 1 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 708  wex 1492  ∃!weu 2026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-eu 2029
This theorem is referenced by:  reuun2  3418
  Copyright terms: Public domain W3C validator