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Theorem alneu 41725
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 5008 . . 3 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
2 exnal 1903 . . 3 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
31, 2sylib 208 . 2 (∃!𝑥𝜑 → ¬ ∀𝑥𝜑)
43con2i 134 1 (∀𝑥𝜑 → ¬ ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1630  wex 1853  ∃!weu 2607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-nul 4941  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-eu 2611  df-mo 2612
This theorem is referenced by:  eu2ndop1stv  41726
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