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Theorem exists2 2597
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 1814 . . . . . 6 (∃!x x = xx∃!x x = x)
2 hba1 1367 . . . . . 6 (xφxxφ)
3 exists1 1849 . . . . . . 7 (∃!x x = xx x = y)
4 ax16 1604 . . . . . . 7 (x x = y → (φxφ))
53, 4sylbi 112 . . . . . 6 (∃!x x = x → (φxφ))
61, 2, 5exlimdh 1418 . . . . 5 (∃!x x = x → (xφxφ))
76com12 25 . . . 4 (xφ → (∃!x x = xxφ))
8 alex 2537 . . . 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 1267  wex 1314   = wceq 1325  ∃!weu 1804
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 1268  ax-7 1269  ax-gen 1270  ax-ie1 1315  ax-ie2 1316  ax-8 1329  ax-10 1330  ax-11 1331  ax-i12 1332  ax-4 1334  ax-17 1352  ax-i9 1356  ax-ial 1361  ax-3 2512
This theorem depends on definitions:  df-bi 108  df-tru 1191  df-fal 1192  df-nf 1282  df-sb 1557  df-eu 1807
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