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Theorem eupick 2208
Description: Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing  x such that 
ph is true, and there is also an  x (actually the same one) such that  ph and  ps are both true, then  ph implies  ps 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  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )

Proof of Theorem eupick
StepHypRef Expression
1 eumo 2185 . 2  |-  ( E! x ph  ->  E* x ph )
2 mopick 2207 . 2  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
31, 2sylan 459 1  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1529   E!weu 2145   E*wmo 2146
This theorem is referenced by:  eupicka  2209  eupickb  2210  reupick  3454  reupick3  3455  copsexg  4252  eusv2nf  4532  reusv2lem3  4537  funssres  5260  tz6.12-1  5505  oprabid  5844  txcn  17315  isch3  21814  copsexgb  24365  iotasbc  27019  bnj849  28225
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150
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