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Theorem euequ1 2292
 Description: Equality has existential uniqueness. Special case of eueq1 3009 proved using only predicate calculus. (Contributed by Stefan Allan, 4-Dec-2008.)
Assertion
Ref Expression
euequ1 ∃!x x = y
Distinct variable group:   x,y

Proof of Theorem euequ1
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 a9ev 1656 . 2 x x = y
2 equtr2 1688 . . 3 ((x = y z = y) → x = z)
32gen2 1547 . 2 xz((x = y z = y) → x = z)
4 equequ1 1684 . . 3 (x = z → (x = yz = y))
54eu4 2243 . 2 (∃!x x = y ↔ (x x = y xz((x = y z = y) → x = z)))
61, 3, 5mpbir2an 886 1 ∃!x x = y
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209 This theorem is referenced by:  copsexg  4607  oprabid  5550  scancan  6331
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