For example, if I say "all dogs are animals" I can symbolize this as "∀x, If $x$ is a dog, then $x$ is an animal", or if I say "every person likes ice cream", I can symbolize it as "∀x, if $x$ is a person, then $x$ likes ice cream". So, can every implication be read like "all $p(x)$, is/are/likes.. $q(x)$"?
Another example: "let $(S,≼)$ be a partially ordered set. If $S$ is well-ordered, then $S$ is totally ordered" can I read this theorem as "every well ordered poset is totally ordered"? which is a little less confusing to understand
Thank you!