1\DOC EXISTS_IMP_CONV 2 3\TYPE {EXISTS_IMP_CONV : conv} 4 5\SYNOPSIS 6Moves an existential quantification inwards through an implication. 7 8\KEYWORDS 9conversion, quantifier, existential, implication. 10 11\DESCRIBE 12When applied to a term of the form {?x. P ==> Q}, where {x} is not free in 13both {P} and {Q}, {EXISTS_IMP_CONV} returns a theorem of one of three forms, 14depending on occurrences of the variable {x} in {P} and {Q}. If {x} is free 15in {P} but not in {Q}, then the theorem: 16{ 17 |- (?x. P ==> Q) = (!x.P) ==> Q 18} 19is returned. If {x} is free in {Q} but not in {P}, then the 20result is: 21{ 22 |- (?x. P ==> Q) = P ==> (?x.Q) 23} 24And if {x} is free in neither {P} nor {Q}, then the result is: 25{ 26 |- (?x. P ==> Q) = (!x.P) ==> (?x.Q) 27} 28 29 30\FAILURE 31{EXISTS_IMP_CONV} fails if it is applied to a term not of the form 32{?x. P ==> Q}, or if it is applied to a term {?x. P ==> Q} in which the 33variable {x} is free in both {P} and {Q}. 34 35\SEEALSO 36Conv.LEFT_IMP_FORALL_CONV, Conv.RIGHT_IMP_EXISTS_CONV. 37\ENDDOC 38