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Theorem eunex 5396
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 dtruALT2 5376 . . . 4 ¬ ∀𝑥 𝑥 = 𝑦
2 albi 1815 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥 𝑥 = 𝑦))
31, 2mtbiri 327 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
43exlimiv 1928 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
5 eu6 2572 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 exnal 1824 . 2 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
74, 5, 63imtr4i 292 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wal 1535  wex 1776  ∃!weu 2566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-nul 5312  ax-pow 5371
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-mo 2538  df-eu 2567
This theorem is referenced by:  reusv2lem2  5405  neutru  36390  alneu  47074
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