Theorem exists2 2133

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		hbeu1 2046	. . . . . 6 ⊢ (∃!𝑥 𝑥 = 𝑥 → ∀𝑥∃!𝑥 𝑥 = 𝑥)
2		hba1 1550	. . . . . 6 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
3		exists1 2132	. . . . . . 7 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
4		ax16 1823	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
5	3, 4	sylbi 121	. . . . . 6 ⊢ (∃!𝑥 𝑥 = 𝑥 → (𝜑 → ∀𝑥𝜑))
6	1, 2, 5	exlimdh 1606	. . . . 5 ⊢ (∃!𝑥 𝑥 = 𝑥 → (∃𝑥𝜑 → ∀𝑥𝜑))
7	6	com12 30	. . . 4 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ∀𝑥𝜑))
8		alexim 1655	. . . 4 ⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
9	7, 8	syl6 33	. . 3 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑))
10	9	con2d 625	. 2 ⊢ (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥))
11	10	imp 124	1 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1361   = wceq 1363  ∃wex 1502  ∃!weu 2036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039

This theorem is referenced by: (None)