Theorem eupick 2098

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 2051	. 2 |- ( E! x ph -> E* x ph )
2		mopick 2097	. 2 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
3	1, 2	sylan 281	1 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1485   E!weu 2019   E*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023

This theorem is referenced by:  eupicka  2099  eupickb  2100  reupick  3411  reupick3  3412  copsexg  4229  eusv2nf  4441  funssres  5240  oprabid  5885  txcn  13069