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Theorem eunex 5278
 Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.) (Proof shortened by BJ, 2-Jan-2023.)
Assertion
Ref Expression
eunex (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Proof of Theorem eunex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dtru 5258 . . . 4 ¬ ∀𝑥 𝑥 = 𝑦
2 albi 1820 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥 𝑥 = 𝑦))
31, 2mtbiri 330 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
43exlimiv 1932 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
5 eu6 2660 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 exnal 1828 . 2 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
74, 5, 63imtr4i 295 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781  ∃!weu 2654 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-nul 5196  ax-pow 5253 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2624  df-eu 2655 This theorem is referenced by:  reusv2lem2  5287  neutru  33815  amosym1  33834  alneu  43610
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