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

Step	Hyp	Ref	Expression
1		eumo 2068	. 2 |- ( E! x ph -> E* x ph )
2		mopick 2114	. 2 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
3	1, 2	sylan 283	1 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )

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