Good Hygiene in using Quantifiers

I think one cannot do better than to quote Halmos on How to write Mathematics (page 142):

The symbolism of formal logic is indispensable in the discussion of the logic of mathematics, but used as a means of transmitting ideas from one mortal to another it becomes a cumbersome code. The author had to code his thoughts in it (I deny that anybody thinks in terms of $\exists$, $\forall$, $\wedge$, and the like), and the reader has to decode what the author wrote; both steps are a waste of time and an obstruction to understanding. Symbolic presentation, in the sense of either the modern logician or the classical epsilontist, is something that machines can write and few but machines can read.

Logical equivalence isn't the issue. The issue is that symbols are hard to read quickly, and using them when they're not necessary slows down the reader. You can find similar advice (avoiding $\forall, \exists$, and so forth) in Knuth, Larrabee, and Roberts' Mathematical Writing on the very first page.

It seems to me that many students think writing using formal symbols makes what they write more rigorous. This is generally not true.