Theorem eupick 2105

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 2058	. 2 |- ( E! x ph -> E* x ph )
2		mopick 2104	. 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 1492   E!weu 2026   E*wmo 2027

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030

This theorem is referenced by:  eupicka  2106  eupickb  2107  reupick  3420  reupick3  3421  copsexg  4245  eusv2nf  4457  funssres  5259  oprabid  5907  txcn  13778