# subst($\Sigma$,s)={subst($\alpha$,s)|$\alpha\in\Sigma$}

I trying to proove this: For a set of verses $\Sigma$ and a function $s:Var\rightarrow WFF$ : Defenition: subst($\Sigma$,s)={subst($\alpha$,s)|$\alpha\in\Sigma$} . Proove that for every $\varphi$ : if $\Sigma\vdash\varphi$ than $subst(\Sigma,s)\vdash subst(\varphi,s)$.

I started from the basic: if $\varphi\in\Sigma$ than $\Sigma\vdash\varphi$ and by defention of subst, $subst(\varphi,s)\in subst(\Sigma,s)$ than $subst(\Sigma,s)\vdash subst(\varphi,s)$. Now, i am trying to find the next untrivial movement to proove it for every $\varphi$ that $\Sigma\vdash\varphi$ but can't see it.

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This notation and terminology is unfamiliar to myself. What is meant by a verse? What is $\mathrm{substr}$ supposed to represent? substitution? Given a verse $\alpha$, how is $\mathrm{substr} ( \alpha , s )$ defined? – arjafi Jan 1 '13 at 13:17
the defenition of subst: for $i\in N$ subst($p_i$,s)=s($p_i$). for $\alpha,\beta\in WFF$ subst($(\neg\alpha)$,s)=$(\neg subst(\alpha,s))$ and $subst((\alpha\circ\beta),s)=(subst(\alpha,s)\circ subst(\beta,s))$ – Michael Cohen Jan 1 '13 at 13:26
While $\circ\in${$\vee,\wedge,\rightarrow$}. And subst(O,s)=O , if $O\in{T,F}$. – Michael Cohen Jan 1 '13 at 13:33

Which deducibility relation (expressed by '$\vdash$') is in question here? After all, we can easily concoct deviant deducibility relations which don't respect substitution. So it matters crucially which deduction system you are working with.
Suppose it is a natural deduction system. Then the individual rules $R$ are usually presented in such a way that if you can readily check that if $\varphi$ follows immediately from assumptions $\Sigma$ by the rule $R$, then $subst(\varphi,s)$ follows from $subst(\Sigma,s)$. And then you can argue by induction on the size of proofs that if $\Sigma \vdash \varphi$ then $subst(\Sigma,s)\vdash subst(\varphi,s)$.