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Theorem eupick 2336
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 2297 . 2  |-  ( E! x ph  ->  E* x ph )
2 mopick 2334 . 2  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
31, 2sylan 473 1  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   E.wex 1659   E!weu 2266   E*wmo 2267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-eu 2270  df-mo 2271
This theorem is referenced by:  eupicka  2337  eupickb  2338  reupick  3763  reupick3  3764  eusv2nf  4623  reusv2lem3  4628  copsexg  4707  funssres  5641  oprabid  6332  txcn  20572  isch3  26729  bnj849  29524  iotasbc  36407
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