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Theorem exists2 1851
 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 1776 . . . . . 6 (∃!x x = xx∃!x x = x)
2 hba1 1364 . . . . . 6 (xφxxφ)
3 exists1 1850 . . . . . . 7 (∃!x x = xx x = y)
4 ax-16 1570 . . . . . . 7 (x x = y → (φxφ))
53, 4sylbi 112 . . . . . 6 (∃!x x = x → (φxφ))
61, 2, 5exlimd 1407 . . . . 5 (∃!x x = x → (xφxφ))
76com12 25 . . . 4 (xφ → (∃!x x = xxφ))
8 alex 1759 . . . 4 (xφ ↔ ¬ x ¬ φ)
97, 8syl6ib 148 . . 3 (xφ → (∃!x x = x → ¬ x ¬ φ))
109con2d 535 . 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 1766 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 526  ax-in2 527  ax-io 606  ax-3 719  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 1349  ax-i9 1353  ax-ial 1358  ax-16 1570 This theorem depends on definitions:  df-bi 108  df-tru 1201  df-fal 1202  df-eu 1770
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