If we have two binary predicates $S(x,y)$ (stands for spouse) and $P(x,y)$ (stands for parent), the formula
1: $$ \forall x_1 \forall x_2 \forall x_3 (S(x_1,x_2) \land P(x_1,x_3) \rightarrow P(x_2,x_3)) $$ means that every one of the spouse's child must also be the other's child (I'm not sure is the formula correct or not).
Then, what would the next two formulas mean
2: $$ \forall x_1 \forall x_2 \exists x_3 (S(x_1,x_2) \land P(x_1,x_3) \rightarrow P(x_2,x_3)) $$ 3: $$ \forall x_1 \forall x_2(S(x_1,x_2) \land P(x_1,x_3) \rightarrow P(x_2,x_3)) $$ apparently, $x_3$ in formula 3 is a free variable, while it's bound in the formula 1 and 2. However, I'm still confused about what the difference is between them, especially between 1 and 3, in my opinions they seem to be the same. How should I recognize the difference in this and other specific formulas between free and bound variables?
Thanks for your answering, if I made some mistakes in my narrative, please correct me.