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Theorem eupick 2115
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 2068 . 2  |-  ( E! x ph  ->  E* x ph )
2 mopick 2114 . 2  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
31, 2sylan 283 1  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   E.wex 1502   E!weu 2036   E*wmo 2037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040
This theorem is referenced by:  eupicka  2116  eupickb  2117  reupick  3431  reupick3  3432  copsexg  4256  eusv2nf  4468  funssres  5270  oprabid  5920  txcn  14128
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