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Theorem euor2 2512
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 2025 . . 3 𝑥𝑥𝜑
21nfn 1782 . 2 𝑥 ¬ ∃𝑥𝜑
3 19.8a 2050 . . . 4 (𝜑 → ∃𝑥𝜑)
43con3i 150 . . 3 (¬ ∃𝑥𝜑 → ¬ 𝜑)
5 biorf 420 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
65bicomd 213 . . 3 𝜑 → ((𝜑𝜓) ↔ 𝜓))
74, 6syl 17 . 2 (¬ ∃𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
82, 7eubid 2486 1 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wex 1702  ∃!weu 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1703  df-nf 1708  df-eu 2472
This theorem is referenced by:  reuun2  3902
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