Theorem exists2 2665

Description: A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 4-Mar-2023.)

Assertion

Ref	Expression
exists2	⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)

Proof of Theorem exists2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axc16nf 2264	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
2	1	nfrd 1789	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 → ∀𝑥𝜑))
3	2	com12 32	. . . 4 ⊢ (∃𝑥𝜑 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝜑))
4		exists1 2664	. . . 4 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
5		alex 1824	. . . . 5 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
6	5	bicomi 224	. . . 4 ⊢ (¬ ∃𝑥 ¬ 𝜑 ↔ ∀𝑥𝜑)
7	3, 4, 6	3imtr4g 296	. . 3 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑))
8	7	con2d 134	. 2 ⊢ (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥))
9	8	imp 406	1 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1535  ∃wex 1777  ∃!weu 2571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-nf 1782  df-mo 2543  df-eu 2572

This theorem is referenced by: (None)