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Theorem alneu 42019
 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 5089 . . 3 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
2 exnal 1925 . . 3 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
31, 2sylib 210 . 2 (∃!𝑥𝜑 → ¬ ∀𝑥𝜑)
43con2i 137 1 (∀𝑥𝜑 → ¬ ∃!𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1654  ∃wex 1878  ∃!weu 2639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-nul 5013  ax-pow 5065 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-mo 2605  df-eu 2640 This theorem is referenced by:  eu2ndop1stv  42020
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