Theorem eunex 5326

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		dtruALT2 5306	. . . 4 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2		albi 1825	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥 𝑥 = 𝑦))
3	1, 2	mtbiri 328	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
4	3	exlimiv 1937	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
5		eu6 2578	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6		exnal 1834	. 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
7	4, 5, 6	3imtr4i 293	1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207  ∀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 5235  ax-pow 5301

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:  reusv2lem2  5335  neutru  36642  alneu  47594