Theorem alneu 44149

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 5257	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
2		exnal 1833	. . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
3	1, 2	sylib 221	. 2 ⊢ (∃!𝑥𝜑 → ¬ ∀𝑥𝜑)
4	3	con2i 141	1 ⊢ (∀𝑥𝜑 → ¬ ∃!𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1786  ∃!weu 2569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-nul 5174  ax-pow 5232

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-mo 2540  df-eu 2570

This theorem is referenced by:  eu2ndop1stv  44150