Consider $W$ is a predicate statement which does not contain any free variable.
$(A)\space\space \exists_x(P(x)\land W) \equiv \exists_xP(x) \land W$
$(B) \space\space \forall_x(P(x)\lor W) \equiv \forall_xP(x) \lor W$
$(C) \space\space\forall_x(P(X) \implies W) \equiv \forall_xP(X) \implies W $
$(D) \space\space\exists_x(P(X) \implies W) \equiv \forall_xP(X) \implies W $
I figured out that option $C$ is not valid logical statement. But my sir have provided the answer as $D$.
My reasoning for option C being not valid:
consider $W$ is $false:$ Now LHS evaluates to false if there is at least one value of $x$ for which $P(x)$ evaluates to true. So in particular when there are some elements in domain for which $P(x)$ evaluates to true and there are some for which it evaluates to $false$ both side of statement does not match.
So, any body confirm whether my reasoning is correct or not(If not please provide reason)? Thanks.