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EXISTS_AND_CONV                                                 (Conv)
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EXISTS_AND_CONV : conv

SYNOPSIS
Moves an existential quantification inwards through a conjunction.

KEYWORDS
conversion, quantifier, existential, conjunction.

DESCRIBE
When applied to a term of the form {?x. P /\ Q}, where {x} is not free in both
{P} and {Q}, {EXISTS_AND_CONV} returns a theorem of one of three forms,
depending on occurrences of the variable {x} in {P} and {Q}.  If {x} is free
in {P} but not in {Q}, then the theorem:

   |- (?x. P /\ Q) = (?x.P) /\ Q

is returned.  If {x} is free in {Q} but not in {P}, then the
result is:

   |- (?x. P /\ Q) = P /\ (?x.Q)

And if {x} is free in neither {P} nor {Q}, then the result is:

   |- (?x. P /\ Q) = (?x.P) /\ (?x.Q)




FAILURE
{EXISTS_AND_CONV} fails if it is applied to a term not of the form
{?x. P /\ Q}, or if it is applied to a term {?x. P /\ Q} in which the
variable {x} is free in both {P} and {Q}.

SEEALSO
Conv.AND_EXISTS_CONV, Conv.EXISTS_AND_REORDER_CONV,
Conv.LEFT_AND_EXISTS_CONV, Conv.RIGHT_AND_EXISTS_CONV.

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