Prove that the following set of formulas is contradictory

Question

Prove that the following set of formulas is contradictory: $$ \left \{ \forall x(Px \implies Qx), \forall x(Qx\implies \exists yRx,y), \exists xPx, \forall x \forall y \neg Rx,y \right \} $$

Answer 1

There exists some $x_0$ such that $Px_0$, which implies $Qx_0$, which implies $(Rx_0,y_0)$ for some $y_0$. But because of $\forall x \forall y \neg (Rx,y)$, we simultaneously have $\neg(Rx_0,y_0)$, so you have a contradiction.