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Theorem eupick 1976
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 1929 . 2 (∃!xφ∃*xφ)
2 mopick 1975 . 2 ((∃*xφ x(φ ψ)) → (φψ))
31, 2sylan 267 1 ((∃!xφ x(φ ψ)) → (φψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wex 1378  ∃!weu 1897  ∃*wmo 1898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by:  eupicka  1977  eupickb  1978  reupick  3215  reupick3  3216  copsexg  3972  eusv2nf  4154  funssres  4885  oprabid  5480
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