Theorem exists2 2711

Description: A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

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		nfeu1 2628	. . . . . 6 ⊢ Ⅎ𝑥∃!𝑥 𝑥 = 𝑥
2		nfa1 2184	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥𝜑
3		exists1 2710	. . . . . . 7 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
4		axc16 2300	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
5	3, 4	sylbi 207	. . . . . 6 ⊢ (∃!𝑥 𝑥 = 𝑥 → (𝜑 → ∀𝑥𝜑))
6	1, 2, 5	exlimd 2243	. . . . 5 ⊢ (∃!𝑥 𝑥 = 𝑥 → (∃𝑥𝜑 → ∀𝑥𝜑))
7	6	com12 32	. . . 4 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ∀𝑥𝜑))
8		alex 1901	. . . 4 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
9	7, 8	syl6ib 241	. . 3 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑))
10	9	con2d 131	. 2 ⊢ (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥))
11	10	imp 393	1 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382  ∀wal 1629  ∃wex 1852  ∃!weu 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-eu 2622

This theorem is referenced by: (None)