Assume that in a formal proof I have
$T \cup \{ \varphi \} \vdash \varphi$
$T \cup \{ \varphi \} \vdash \lnot \varphi$
Question 1: can I then deduce $T \cup \{ \varphi \} \vdash \lnot \varphi \land \varphi$? I think there should be a rule of deduction that tells me that I can do that but there is no such rule in my lecture notes. What I do have is the following:
$\{ \psi , \varphi \} \vdash \psi \land \varphi$
So I guess my question boils down to the following:
If I have $T \vdash \varphi$, can I do $T \cup \{ \varphi \}\vdash $?
Question 2: is similar. If I have
$T \cup \{ \varphi \} \vdash \varphi \land \lnot \varphi$ ,can I deduce $T \cup \{ \varphi \} \vdash \lnot \varphi$?
Many thanks for your help.