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

Theorem euor2 1974
Description: Introduce or eliminate a disjunct in a uniqueness 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 1400 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
21hbn 1560 . 2 (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)
3 19.8a 1498 . . . 4 (𝜑 → ∃𝑥𝜑)
43con3i 572 . . 3 (¬ ∃𝑥𝜑 → ¬ 𝜑)
5 orel1 654 . . . 4 𝜑 → ((𝜑𝜓) → 𝜓))
6 olc 642 . . . 4 (𝜓 → (𝜑𝜓))
75, 6impbid1 134 . . 3 𝜑 → ((𝜑𝜓) ↔ 𝜓))
84, 7syl 14 . 2 (¬ ∃𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
92, 8eubidh 1922 1 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  wex 1397  ∃!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-eu 1919
This theorem is referenced by:  reuun2  3248
  Copyright terms: Public domain W3C validator