Notation matter concerning 'or'-elimination

Question

I have to show that $\{(\phi\lor\psi),(\lnot\phi)\}\vdash\psi$ using the following natural deduction rule:

I don't know which of these is correct in term of notation:

Could you please tell me? Thank you in advance!

Answer 1

You wish to prove $\dfrac{(\phi\vee\psi)\quad (\neg \phi)}{\psi}$ using $\require{cancel}\dfrac{(\phi^1\vee \psi^2)\quad \dfrac{(\cancelto{1}[color=gray]{\color{black}\phi}\wedge\neg\phi)}{\psi}{\small ?}\quad \dfrac{(\cancelto{2}[color=gray]{\color{black}\psi}\wedge\neg\phi)}{\psi}{\small ?}}{\psi}{\small\vee\mathsf E}$

Clearly the second justification is conjunction elimination, where as the first involves a negation elimination.

$$\require{cancel}\dfrac{(\phi^1\vee \psi^2)\quad \dfrac{\dfrac{(\cancelto{1}[color=gray]{\color{black}\phi}\wedge\neg\phi)}{\bot}{\small\neg\mathsf E}}{\psi}{\small\sf RAA}\quad \dfrac{(\cancelto{2}[color=gray]{\color{black}\psi}\wedge\neg\phi)}{\psi}{\small\wedge\mathsf E}}{\psi}{\small\vee\mathsf E}$$