Depends on which President I suppose, Obama pulled money out of his arse I guess because he spent plenty and I don't recall any spending bill that covered his spending sprees..
I believe you are right.
8 SEC. 6. AVAILABILITY OF FUNDS.
9 Each amount designated in this Act by the Congress
10 for Overseas Contingency Operations/Global War on Ter11
rorism pursuant to section 251(b)(2)(A)(ii) of the Bal12
anced Budget and Emergency Deficit Control Act of 1985
13 shall be available (or rescinded, if applicable) only if the
14 President subsequently so designates all such amounts
15 and transmits such designations to the Congress.