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.