Theorem eupick 2093

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 2046	. 2 |- ( E! x ph -> E* x ph )
2		mopick 2092	. 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 1480   E!weu 2014   E*wmo 2015

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018

This theorem is referenced by:  eupicka  2094  eupickb  2095  reupick  3406  reupick3  3407  copsexg  4222  eusv2nf  4434  funssres  5230  oprabid  5874  txcn  12915