Theorem eunex 5281

Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.) (Proof shortened by BJ, 2-Jan-2023.)

Assertion

Ref	Expression
eunex	⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Proof of Theorem eunex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dtru 5262	. . . 4 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2		albi 1813	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥 𝑥 = 𝑦))
3	1, 2	mtbiri 329	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
4	3	exlimiv 1925	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
5		eu6 2653	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6		exnal 1821	. 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
7	4, 5, 6	3imtr4i 294	1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1529  ∃wex 1774  ∃!weu 2647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2170  ax-nul 5201  ax-pow 5257

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-mo 2616  df-eu 2648

This theorem is referenced by:  reusv2lem2  5290  neutru  33748  amosym1  33767  alneu  43313