Consider the following two propositions:
$$S_1 \subseteq S_2 := \forall x. x \in S_1 \implies x \in S_2$$
$$S_1 \subseteq S_2 := \forall x \in S_1. x \in S_2$$
These two expressions are used to denote the same thing and both often used to define the subset relation. But why is that so? Can the equivalence of these two expressions be demonstrated or is it just convention? Intuitively I would not agree that these say the same thing. $x$ being a member of $S_1$ does not in itself imply that $x$ is also a member of $S_2$. The second definition makes no mention of a logical consequence of $x$ necessarily having to be a member of $S_2$ when it is a member of $S_1$. It merely states that if both of these two things are the case, then $S_1$ is a subset of $S_2$.
So what justifies the implication and what makes these two statements equivalent in particular and what makes other pairs of statements that might be analogous (one involving an implication the other a restriction of the quantifier) equivalent, respectively?
By the usual conventions (at least in axiomatic set theory), these two statements are the same by definition. More precisely, the notation $$\forall x\in S. \varphi$$ is defined to be an abbreviation for $$\forall x.(x\in S\Rightarrow \varphi).$$ So your second statement is defined to mean your first statement.