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

Theorem euor2 2690
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.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
Assertion
Ref Expression
euor2 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))

Proof of Theorem euor2
StepHypRef Expression
1 nfe1 2145 . . 3 𝑥𝑥𝜑
21nfn 1848 . 2 𝑥 ¬ ∃𝑥𝜑
3 19.8a 2170 . . . 4 (𝜑 → ∃𝑥𝜑)
43con3i 157 . . 3 (¬ ∃𝑥𝜑 → ¬ 𝜑)
5 biorf 930 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
65bicomd 224 . . 3 𝜑 → ((𝜑𝜓) ↔ 𝜓))
74, 6syl 17 . 2 (¬ ∃𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
82, 7eubid 2666 1 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wo 841  wex 1771  ∃!weu 2646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-mo 2615  df-eu 2647
This theorem is referenced by:  reuun2  4283
  Copyright terms: Public domain W3C validator