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Theorem exists2 1773
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 1698 . . . . . 6 (∃!x x = xx∃!x x = x)
2 hba1 1316 . . . . . 6 (xφxxφ)
3 exists1 1772 . . . . . . 7 (∃!x x = xx x = y)
4 ax-16 1502 . . . . . . 7 (x x = y → (φxφ))
53, 4sylbi 113 . . . . . 6 (∃!x x = x → (φxφ))
61, 2, 5exlimd 1351 . . . . 5 (∃!x x = x → (xφxφ))
76com12 26 . . . 4 (xφ → (∃!x x = xxφ))
8 alex 1684 . . . 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 1231  wex 1270   = wceq 1279  ∃!weu 1688
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 1232  ax-7 1234  ax-gen 1235  ax-ie1 1271  ax-ie2 1272  ax-8 1283  ax-10 1284  ax-11 1285  ax-i12 1287  ax-4 1288  ax-17 1297  ax-i9 1299  ax-ial 1310  ax-16 1502
This theorem depends on definitions:  df-bi 109  df-tru 1209  df-fal 1210  df-eu 1692
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