Theorem eupick

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)

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

Proof

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