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Theorem eupick 2355
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 2315 . 2  |-  ( E! x ph  ->  E* x ph )
2 mopick 2353 . 2  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
31, 2sylan 469 1  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   E.wex 1617   E!weu 2284   E*wmo 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-eu 2288  df-mo 2289
This theorem is referenced by:  eupicka  2356  eupickb  2357  reupick  3779  reupick3  3780  eusv2nf  4635  reusv2lem3  4640  copsexg  4722  funssres  5610  oprabid  6297  txcn  20296  isch3  26360  iotasbc  31570  bnj849  34403
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