Clarification of a proof as written in Margaris's book In Angelo Margaris's book First Order Mathematical Logic it is written (see below),




Then,



Questions


*

*In the above proof I don't understand what am I supposed to do at the second step. Can anyone explain that to me?

*How from "$P$ admits $t$ for $v$" it follows that "$\sim P$ admits $t$ for $v$" (because otherwise you can't use Axiom 5 which says that $\forall v Q\to Q(t/v)$ provides $Q$ admits $t$ for $v$)?

Answers


*

*In the second step "SC,1" means that the following, $$(\forall v{\sim}P\to{\sim}P(t/v))\to (P(t/v)\to{\sim}\forall v{\sim}P)$$ is a tautology.

*Hint: If ${\sim}P$ doesn't admit $t$ for $v$, then there exists at least one variable $u$ in $t$ such that it occurs in a subformula of the form $\mathtt{Q}uM$ of ${\sim}P$ where $\mathtt{Q}$ is a quantifier and $v$ is free in $M$ (why?).   
