Theorem eupick 2267

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!xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Proof of Theorem eupick

Step	Hyp	Ref	Expression
1		eumo 2244	. 2 |- (E!xph -> E*xph)
2		mopick 2266	. 2 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
3	1, 2	sylan 457	1 |- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541  E!weu 2204  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  eupicka  2268  eupickb  2269  reupick  3540  reupick3  3541  copsexg  4608  funssres  5145  oprabid  5551