The English word ANY as in ANYTHING or ANYONE can be very tricky.

Generally this word means the same as ALL. For example, Any student who studies will pass this course or (x)(Sx ⊃ Px)

But after the word IF it seems to mean SOME. For example, If any student studies, Arlene does or (∃x)Sx ⊃ Sa

The translation is logically equivalent to (x)(Sx ⊃ Sa) which does preserve the original insight about ANY. (Note carefully that the quantifiers have different scopes.)

The IF ANY... construction canNOT always be translated with  (∃x)... ⊃  where the  ⊃  is the main operator.

Consider: If any child has a dog, then it is a pet. The grammar and the comma make it look as if the main operation in the English is the IF...THEN but this is deceiving. If the deception is followed the INCORRECT translation becomes

  (∃x)[Cx • (∃y)(Dy • Hxy)] ⊃ Py but this leaves the y in Py free.

The difficulty occurs because the pronoun IT makes a reference back across the THEN to DOG. So the quantifier which binds the individuals which are dogs to the IT has wider scope than the IF...THEN. Correct translations are

(x){Cx ⊃ (y)[(Dy • Hxy) ⊃ Py]} and (x)(y){[Cx • (Dy • Hxy)] ⊃ Py}