  In functional analysis it is of interest to study the following general
question:
  Is the uniform version of a property that holds in all Banach spaces also
valid in all Banach spaces?
  Examples of affirmative answers to the above question are the host of proofs
of almost-isometric versions of well known isometric theorems. Another example
is Rosenthal's uniform version of Krivine's Theorem. Using an extended version
of Henson's Compactness result for positive bounded formulas in normed
structures, we show that the answer of the above question is in fact yes for
every property that can be expressed in a particular infinitary language.
