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Theorem alneu 44616
Description: If a statement holds for all sets, there is not a unique set for which the statement holds. (Contributed by Alexander van der Vekens, 28-Nov-2017.)
Assertion
Ref Expression
alneu (∀𝑥𝜑 → ¬ ∃!𝑥𝜑)

Proof of Theorem alneu
StepHypRef Expression
1 eunex 5313 . . 3 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
2 exnal 1829 . . 3 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
31, 2sylib 217 . 2 (∃!𝑥𝜑 → ¬ ∀𝑥𝜑)
43con2i 139 1 (∀𝑥𝜑 → ¬ ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1537  wex 1782  ∃!weu 2568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-nul 5230  ax-pow 5288
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-mo 2540  df-eu 2569
This theorem is referenced by:  eu2ndop1stv  44617
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