More specifically, going up a fifth augments the fourth by a semitone to become the seventh and going down a fifth diminishes the seventh by a semitone to become the fourth. In your example, F the fourth became F# the seventh and B the seventh became Bb the fourth. A fifth away _IS_ a semitone away.

Defining the following eight functions:

  SEMITONE =  +1 mod 12;  ANTI_SEMITONE = SEVENTH;
  FOURTH   =  +5 mod 12;  ANTI_FOURTH   = FIFTH;
  FIFTH    =  +7 mod 12;  ANTI_FIFTH    = FOURTH;
  SEVENTH  = +11 mod 12;  ANTI_SEVENTH  = SEMITONE;

We ask ourselves why the following holds:

   FOURTH(X) = ANTI_SEMITONE(SEVENTH(FIFTH(X)))
  SEVENTH(X) = SEMITONE(FOURTH(ANTI_FIFTH(X)))

Substituting, the answer is clear:

  X +  5 = X + 7 + 11 - 1
  X + 11 = X - 7 +  5 + 1

Keys one fifth apart differ by one note because:

  1 = 11 + 7 - 5 mod 12

Algebraically it seems to work out...