Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  euor2 Structured version   Visualization version   GIF version

Theorem euor2 2502
 Description: Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
Assertion
Ref Expression
euor2 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))

Proof of Theorem euor2
StepHypRef Expression
1 nfe1 2014 . . 3 𝑥𝑥𝜑
21nfn 1768 . 2 𝑥 ¬ ∃𝑥𝜑
3 19.8a 2039 . . . 4 (𝜑 → ∃𝑥𝜑)
43con3i 149 . . 3 (¬ ∃𝑥𝜑 → ¬ 𝜑)
5 biorf 419 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
65bicomd 212 . . 3 𝜑 → ((𝜑𝜓) ↔ 𝜓))
74, 6syl 17 . 2 (¬ ∃𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
82, 7eubid 2476 1 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382  ∃wex 1695  ∃!weu 2458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701  df-eu 2462 This theorem is referenced by:  reuun2  3869
 Copyright terms: Public domain W3C validator