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Theorem eupick 2267
 Description: Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing x such that φ is true, and there is also an x (actually the same one) such that φ and ψ are both true, then φ implies ψ regardless of x. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)
Assertion
Ref Expression
eupick ((∃!xφ x(φ ψ)) → (φψ))

Proof of Theorem eupick
StepHypRef Expression
1 eumo 2244 . 2 (∃!xφ∃*xφ)
2 mopick 2266 . 2 ((∃*xφ x(φ ψ)) → (φψ))
31, 2sylan 457 1 ((∃!xφ x(φ ψ)) → (φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205 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-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:  eupicka  2268  eupickb  2269  reupick  3539  reupick3  3540  copsexg  4607  funssres  5144  oprabid  5550
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