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Theorem exists2 2748
 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.
StepHypRef Expression
1 axc16nf 2257 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
21nfrd 1785 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 → ∀𝑥𝜑))
32com12 32 . . . 4 (∃𝑥𝜑 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝜑))
4 exists1 2747 . . . 4 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
5 alex 1819 . . . . 5 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
65bicomi 225 . . . 4 (¬ ∃𝑥 ¬ 𝜑 ↔ ∀𝑥𝜑)
73, 4, 63imtr4g 297 . . 3 (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑))
87con2d 136 . 2 (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥))
98imp 407 1 ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1528  ∃wex 1773  ∃!weu 2650 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-12 2169 This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1533  df-ex 1774  df-nf 1778  df-mo 2619  df-eu 2651 This theorem is referenced by: (None)
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