# Section 2.4: Problem 14 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Show that $\vdash(\forall x((\neg Px)\rightarrow Qx)\rightarrow\forall y((\neg Qy)\rightarrow Py))$ .
Proof:
1. $\vdash((\neg Px)\rightarrow Qx)\rightarrow((\neg Qx)\rightarrow Px)$ , A1.
2. $\vdash\forall x[((\neg Px)\rightarrow Qx)\rightarrow((\neg Qx)\rightarrow Px)]$ , G:1.
3. $\vdash\forall x[((\neg Px)\rightarrow Qx)\rightarrow((\neg Qx)\rightarrow Px)]\rightarrow\forall x((\neg Px)\rightarrow Qx)\rightarrow\forall x((\neg Qx)\rightarrow Px)$ , A3.
4. $\vdash\forall x((\neg Px)\rightarrow Qx)\rightarrow\forall x((\neg Qx)\rightarrow Px)$ , MP:2+3.
5. $\forall x((\neg Px)\rightarrow Qx)\vdash\forall x((\neg Qx)\rightarrow Px)$ , MP:4.
6. $\vdash\forall x((\neg Qx)\rightarrow Px)\rightarrow((\neg Qy)\rightarrow Py)$ , A2.
7. $\forall x((\neg Px)\rightarrow Qx)\vdash((\neg Qy)\rightarrow Py)$ , MP:5+6.
8. $\forall x((\neg Px)\rightarrow Qx)\vdash\forall y((\neg Qy)\rightarrow Py)$ , G:7.
9. $\vdash(\forall x((\neg Px)\rightarrow Qx)\rightarrow\forall y((\neg Qy)\rightarrow Py))$ , D:8.