\DOC AND_EXISTS_CONV \TYPE {AND_EXISTS_CONV : conv} \SYNOPSIS Moves an existential quantification outwards through a conjunction. \KEYWORDS conversion, quantifier. \DESCRIBE When applied to a term of the form {(?x.P) /\ (?x.Q)}, where {x} is free in neither {P} nor {Q}, {AND_EXISTS_CONV} returns the theorem: { |- (?x. P) /\ (?x. Q) = (?x. P /\ Q) } \FAILURE {AND_EXISTS_CONV} fails if it is applied to a term not of the form {(?x.P) /\ (?x.Q)}, or if it is applied to a term {(?x.P) /\ (?x.Q)} in which the variable {x} is free in either {P} or {Q}. \COMMENTS It may be easier to use higher order rewriting with some of {BOTH_EXISTS_AND_THM}, {LEFT_EXISTS_AND_THM}, and {RIGHT_EXISTS_AND_THM}. \SEEALSO Conv.EXISTS_AND_CONV, Conv.LEFT_AND_EXISTS_CONV, Conv.RIGHT_AND_EXISTS_CONV, BOTH_EXISTS_AND_THM, LEFT_EXISTS_AND_THM, RIGHT_EXISTS_AND_THM. \ENDDOC