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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.
StepHypRef Expression
1 nfeu1 2628 . . . . . 6 𝑥∃!𝑥 𝑥 = 𝑥
2 nfa1 2184 . . . . . 6 𝑥𝑥𝜑
3 exists1 2710 . . . . . . 7 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
4 axc16 2300 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
53, 4sylbi 207 . . . . . 6 (∃!𝑥 𝑥 = 𝑥 → (𝜑 → ∀𝑥𝜑))
61, 2, 5exlimd 2243 . . . . 5 (∃!𝑥 𝑥 = 𝑥 → (∃𝑥𝜑 → ∀𝑥𝜑))
76com12 32 . . . 4 (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ∀𝑥𝜑))
8 alex 1901 . . . 4 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
97, 8syl6ib 241 . . 3 (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑))
109con2d 131 . 2 (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥))
1110imp 393 1 ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)
Colors of variables: wff setvar class
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)
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