It doesn't need to. That much is logically presumed and inherent to any act of Congress, which by its own nature must be legislative.
Where does it explicitly say that only Congress can suspend it?
Article I, Section 1, which governs all the clauses below it in Article I, among them the habeas corpus clause. To deny this is to deny both the plainly stated words of the constitution and the plainly stated understanding of those words given by the founding fathers who witnessed, drafted, and implemented that constitution.
Put differently, you are skating on very thin legal ice, non-seq. Your arguments on habeas corpus are of the exact same character as the "meaning of is" and the "definition of what constitutes sex" in a Clinton deposition; that is to say they reek of orchestrated semantical deception and leave behind them a trail of slime indicative of their intentionally slippery nature.
Ah, so we are talking implied powers. In Article I, the Congress is explicitly mentioned on only 8 clauses but you 'logically presume' that it applies to the remaining clauses as well. Should we 'logically presume' that a state could enter into a treaty, alliance, or confederation; grant Letters of Marque and Reprisal; coin money; emit bills of credit; make anything but gold and silver coin a legal tender; pass a Bill of Attainder, ex post facto law, or law impairing the obligation of contracts; or grant a title of nobility if Congress passed a law allowing them to?
But while we're on the subject, let me ask you a question on an slightly unrelated matter. Since Article IV, Section 3 states that Congressional approval is required for admitting new states and for any change in status such as combining, splitting into two or more states, etc. then couldn't we 'logically presume' that congressional approval would be needed for leaving the Union altogether? I already know the answer to this but it should be amusing to see your explanation. Have at it.
By your definition. It's interesting to note that the founders found the need to constantly state that congressional approval was needed for this and that legislation was needed for that all through Article I. Why is that. Do you suppose that they didn't realize that Article I, Section 1 governed all clauses? At least, according to you and your 'logical presumptions'.
Your arguments on habeas corpus are of the exact same character as the "meaning of is" and the "definition of what constitutes sex" in a Clinton deposition...
Are you suggesting my arguements are similar to your definition of tu quoque? That it means whatever I want it to mean at the time?