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Theorem exists2 2100
 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.
StepHypRef Expression
1 hbeu1 2013 . . . . . 6 (∃!𝑥 𝑥 = 𝑥 → ∀𝑥∃!𝑥 𝑥 = 𝑥)
2 hba1 1517 . . . . . 6 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
3 exists1 2099 . . . . . . 7 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
4 ax16 1790 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
53, 4sylbi 120 . . . . . 6 (∃!𝑥 𝑥 = 𝑥 → (𝜑 → ∀𝑥𝜑))
61, 2, 5exlimdh 1573 . . . . 5 (∃!𝑥 𝑥 = 𝑥 → (∃𝑥𝜑 → ∀𝑥𝜑))
76com12 30 . . . 4 (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ∀𝑥𝜑))
8 alexim 1622 . . . 4 (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
97, 8syl6 33 . . 3 (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑))
109con2d 614 . 2 (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥))
1110imp 123 1 ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332  ∃wex 1469  ∃!weu 2003 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006 This theorem is referenced by: (None)
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