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Theorem exists2 2172
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 1786 . . . . . 6 (∃!x x = xx∃!x x = x)
2 hba1 1365 . . . . . 6 (xφxxφ)
3 exists1 2171 . . . . . . 7 (∃!x x = xx x = y)
4 ax16 1580 . . . . . . 7 (x x = y → (φxφ))
53, 4sylbi 112 . . . . . 6 (∃!x x = x → (φxφ))
61, 2, 5exlimd 1411 . . . . 5 (∃!x x = x → (xφxφ))
76com12 25 . . . 4 (xφ → (∃!x x = xxφ))
8 alex 2087 . . . 4 (xφ ↔ ¬ x ¬ φ)
97, 8syl6ib 148 . . 3 (xφ → (∃!x x = x → ¬ x ¬ φ))
109con2d 537 . 2 (xφ → (x ¬ φ → ¬ ∃!x x = x))
1110imp 113 1 ((xφ x ¬ φ) → ¬ ∃!x x = x)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 95  wal 1266  wex 1313   = wceq 1324  ∃!weu 1777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-in1 527  ax-in2 528  ax-io 609  ax-5 1267  ax-7 1268  ax-gen 1269  ax-ie1 1314  ax-ie2 1315  ax-8 1328  ax-10 1329  ax-11 1330  ax-i12 1331  ax-4 1333  ax-17 1350  ax-i9 1354  ax-ial 1359  ax-3 2062
This theorem depends on definitions:  df-bi 108  df-tru 1190  df-fal 1191  df-sb 1533  df-eu 1780
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