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Theorem eunex 5139
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 5120 . . . 4 ¬ ∀𝑥 𝑥 = 𝑦
2 albi 1782 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥 𝑥 = 𝑦))
31, 2mtbiri 319 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
43exlimiv 1890 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
5 eu6 2591 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 exnal 1790 . 2 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
74, 5, 63imtr4i 284 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wal 1506  wex 1743  ∃!weu 2584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-12 2107  ax-nul 5063  ax-pow 5115
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-mo 2548  df-eu 2585
This theorem is referenced by:  reusv2lem2  5149  neutru  33313  amosym1  33331  alneu  42761
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