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Theorem euex 2227
 Description: Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
euex (∃!xφxφ)

Proof of Theorem euex
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . 3 yφ
21eu1 2225 . 2 (∃!xφx(φ y([y / x]φx = y)))
3 exsimpl 1592 . 2 (x(φ y([y / x]φx = y)) → xφ)
42, 3sylbi 187 1 (∃!xφxφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648  ∃!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 This theorem is referenced by:  eu2  2229  exmoeu  2246  eupickbi  2270  2eu2ex  2278  2exeu  2281  euxfr  3022  fvprc  5325  tz6.12c  5347  ndmfv  5349  dff3  5420  fnoprabg  5585
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