Theorem eupick 2095

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.

Assertion

Ref	Expression
eupick	|- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Proof of Theorem eupick

Step	Hyp	Ref	Expression
1		eumo 2072	. 2 |- (E!xph -> E*xph)
2		mopick 2094	. 2 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
3	1, 2	sylan 597	1 |- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Syntax hints:   -> wi 3   /\ wa 337  E.wex 1615  E!weu 2037  E*wmo 2038

This theorem is referenced by:  eupicka 2096  eupickb 2097  reupick 3081  copsexg 3700  funssres 4562  tz6.12-1 4782  oprabid 5002  chcmhi 11580  iotasbc 17207

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-ex 1616  df-sb 1816  df-eu 2041  df-mo 2042

Copyright terms: Public domain