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Theorem exists2 2045
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 1958 . . . . . 6 (∃!𝑥 𝑥 = 𝑥 → ∀𝑥∃!𝑥 𝑥 = 𝑥)
2 hba1 1478 . . . . . 6 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
3 exists1 2044 . . . . . . 7 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
4 ax16 1741 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
53, 4sylbi 119 . . . . . 6 (∃!𝑥 𝑥 = 𝑥 → (𝜑 → ∀𝑥𝜑))
61, 2, 5exlimdh 1532 . . . . 5 (∃!𝑥 𝑥 = 𝑥 → (∃𝑥𝜑 → ∀𝑥𝜑))
76com12 30 . . . 4 (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ∀𝑥𝜑))
8 alexim 1581 . . . 4 (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
97, 8syl6 33 . . 3 (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑))
109con2d 589 . 2 (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥))
1110imp 122 1 ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1287   = wceq 1289  wex 1426  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951
This theorem is referenced by: (None)
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