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Theorem eupick 2052
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 2005 . 2  |-  ( E! x ph  ->  E* x ph )
2 mopick 2051 . 2  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
31, 2sylan 279 1  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1449   E!weu 1973   E*wmo 1974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496
This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977
This theorem is referenced by:  eupicka  2053  eupickb  2054  reupick  3324  reupick3  3325  copsexg  4124  eusv2nf  4335  funssres  5121  oprabid  5755  txcn  12280
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