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Theorem exists2 2294
 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 nfeu1 2214 . . . . . 6 x∃!x x = x
2 nfa1 1788 . . . . . 6 xxφ
3 exists1 2293 . . . . . . 7 (∃!x x = xx x = y)
4 ax16 2045 . . . . . . 7 (x x = y → (φxφ))
53, 4sylbi 187 . . . . . 6 (∃!x x = x → (φxφ))
61, 2, 5exlimd 1806 . . . . 5 (∃!x x = x → (xφxφ))
76com12 27 . . . 4 (xφ → (∃!x x = xxφ))
8 alex 1572 . . . 4 (xφ ↔ ¬ x ¬ φ)
97, 8syl6ib 217 . . 3 (xφ → (∃!x x = x → ¬ x ¬ φ))
109con2d 107 . 2 (xφ → (x ¬ φ → ¬ ∃!x x = x))
1110imp 418 1 ((xφ x ¬ φ) → ¬ ∃!x x = x)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃!weu 2204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208 This theorem is referenced by: (None)
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