Theorem eupick 2020

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 1973	. 2 |- ( E! x ph -> E* x ph )
2		mopick 2019	. 2 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
3	1, 2	sylan 277	1 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   E.wex 1421   E!weu 1941   E*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  eupicka  2021  eupickb  2022  reupick  3248  reupick3  3249  copsexg  3999  eusv2nf  4206  funssres  4962  oprabid  5557