Theorem eunex 4859

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

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 4857	. . . . 5 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2		alim 1738	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∀𝑥𝜑 → ∀𝑥 𝑥 = 𝑦))
3	1, 2	mtoi 190	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
4	3	exlimiv 1858	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
5	4	adantl 482	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → ¬ ∀𝑥𝜑)
6		eu3v 2498	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
7		exnal 1754	. 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
8	5, 6, 7	3imtr4i 281	1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704  ∃!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  reusv2lem2  4869  reusv2lem2OLD  4870  unnt  32407  amosym1  32425  alneu  41201