Theorem alneu 43330

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

Step	Hyp	Ref	Expression
1		eunex 5293	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
2		exnal 1827	. . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
3	1, 2	sylib 220	. 2 ⊢ (∃!𝑥𝜑 → ¬ ∀𝑥𝜑)
4	3	con2i 141	1 ⊢ (∀𝑥𝜑 → ¬ ∃!𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  ∃wex 1780  ∃!weu 2653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-nul 5212  ax-pow 5268

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-mo 2622  df-eu 2654

This theorem is referenced by:  eu2ndop1stv  43331