Theorem alneu 47587

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 5319	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
2		exnal 1834	. . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
3	1, 2	sylib 219	. 2 ⊢ (∃!𝑥𝜑 → ¬ ∀𝑥𝜑)
4	3	con2i 139	1 ⊢ (∀𝑥𝜑 → ¬ ∃!𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1545  ∃wex 1786  ∃!weu 2572

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-nul 5228  ax-pow 5294

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791  df-mo 2543  df-eu 2573

This theorem is referenced by:  eu2ndop1stv  47588