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Theorem euor2 2665
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 2194 . . 3 𝑥𝑥𝜑
21nfn 1954 . 2 𝑥 ¬ ∃𝑥𝜑
3 19.8a 2216 . . . 4 (𝜑 → ∃𝑥𝜑)
43con3i 152 . . 3 (¬ ∃𝑥𝜑 → ¬ 𝜑)
5 biorf 961 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
65bicomd 215 . . 3 𝜑 → ((𝜑𝜓) ↔ 𝜓))
74, 6syl 17 . 2 (¬ ∃𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
82, 7eubid 2625 1 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wo 874  wex 1875  ∃!weu 2606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-nf 1880  df-mo 2590  df-eu 2607
This theorem is referenced by:  reuun2  4108
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