With these hypotheses:
“The dog is not sleeping and the llama is named Pebbles.”
“We will drink coffee only if the dog is sleeping.”  (See below for the only if )
“If we do not drink coffee, then we will sniff whiteboard markers.”
“If we sniff whiteboard markers, then we become math professors.”


Prove that:
“we become math professors.”




Aside:
Consider the statement "The robot dances only if you feed it oil."  My brain rearranges this to:  "No oil?  Then no dance!" 

In exactly what cases is "The robot dances only if you feed it oil." a false statement?

Robot is dancing	Robot drank oil	Statement is true?
False	False	True --no oil, no dance.
False	True	True -- has oil, but might need a charged battery too.
True	False	False -- no oil, no dance!
True	True	True

To continue with my initial thought, "The robot dances only if you feed it oil" =  "No oil?  Then no dance!"  In a complete sentence. . .

                                                                                                                              =   "If you don't feed it oil then it won't dance."



We've just figured out that "p only if q" is equivalent to "not q implies not p".  This is the contrapositive  of "p implies q", and contrapositive statements are equivalent.


The whole point of this aside.  "p only if q" = "if p then q" = "p implies q".