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Theorem exists2 1775
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 1700 . . . . . 6 (∃!x x = xx∃!x x = x)
2 hba1 1339 . . . . . 6 (xφxxφ)
3 exists1 1774 . . . . . . 7 (∃!x x = xx x = y)
4 ax-16 1523 . . . . . . 7 (x x = y → (φxφ))
53, 4sylbi 113 . . . . . 6 (∃!x x = x → (φxφ))
61, 2, 5exlimd 1375 . . . . 5 (∃!x x = x → (xφxφ))
76com12 26 . . . 4 (xφ → (∃!x x = xxφ))
8 alex 1689 . . . 4 (xφ ↔ ¬ x ¬ φ)
97, 8syl6ib 149 . . 3 (xφ → (∃!x x = x → ¬ x ¬ φ))
109con2d 535 . 2 (xφ → (x ¬ φ → ¬ ∃!x x = x))
1110imp 114 1 ((xφ x ¬ φ) → ¬ ∃!x x = x)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 96  wal 1253  wex 1292   = wceq 1301  ∃!weu 1690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ia1 98  ax-ia2 99  ax-ia3 100  ax-in1 526  ax-in2 527  ax-io 606  ax-5 1254  ax-7 1256  ax-gen 1257  ax-ie1 1293  ax-ie2 1294  ax-8 1305  ax-10 1306  ax-11 1307  ax-i12 1309  ax-4 1310  ax-17 1319  ax-i9 1321  ax-ial 1333  ax-16 1523
This theorem depends on definitions:  df-bi 109  df-tru 1231  df-fal 1232  df-eu 1694
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