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Theorem alneu 43308
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 5281 . . 3 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
2 exnal 1820 . . 3 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
31, 2sylib 220 . 2 (∃!𝑥𝜑 → ¬ ∀𝑥𝜑)
43con2i 141 1 (∀𝑥𝜑 → ¬ ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1528  wex 1773  ∃!weu 2647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2169  ax-nul 5201  ax-pow 5257
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-mo 2616  df-eu 2648
This theorem is referenced by:  eu2ndop1stv  43309
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