Natural Deduction Proof: {A v B, ¬A v C} ⊢ B v C Prove using natural deduction: $(A \lor B), (\lnot A \lor C) ⊢ (B \lor C)$.
My work so far:
1     A v B
2     A v C
3      A      :AS
4       ¬A      :AS
5       _|_     :¬E 3,4
6    ¬¬A      :¬I 4-5
7  --
8  ¬A     :AS

This is the work that I have come up with, for the past 5 hours. Can you please help me?
You are setting up for disjunction elimination, so when you derive contradiction, use explosion to derive the target conclusion.$$\def\fitch#1#2{~~\begin{array}{|l}#1\\\hline#2\end{array}}\fitch{~~1.~A\lor B\hspace{10ex}\textsf{Premise}\\~~2.~\lnot A\lor C\hspace{8.5ex}\textsf{Premise}}{\fitch{~~3.~A\hspace{12.5ex}\textsf{Assumption}}{\fitch{~~4.~\lnot A\hspace{8.5ex}\textsf{Assumption}}{~~5.~\bot\hspace{10ex}\textsf{Negation Elimination}\\~~6.~B\lor C\hspace{5.75ex}\textsf{Explosion (aka }\textit{Ex Falso Quodlibet}\textsf{)}}\\~~\vdots\\~~m.~B\lor C }\\~~\vdots\\~n.~B\lor C}$$
I am sure you can fill out the rest of this &quot;proof by cases&quot;.
You are setting up for disjunction elimination, so when you derive contradiction, use explosion to derive the target conclusion.$$\def\fitch#1#2{~~\begin{array}{|l}#1\\\hline#2\end{array}}\fitch{~~1.~A\lor B\hspace{10ex}\textsf{Premise}\\~~2.~\lnot A\lor C\hspace{8.5ex}\textsf{Premise}}{\fitch{~~3.~A\hspace{12.5ex}\textsf{Assumption}}{\fitch{~~4.~\lnot A\hspace{8.5ex}\textsf{Assumption}}{~~5.~\bot\hspace{10ex}\textsf{Negation Elimination}\\~~6.~B\lor C\hspace{5.75ex}\textsf{Explosion (aka }\textit{Ex Falso Quodlibet}\textsf{)}}\\~~\vdots\\~~m.~B\lor C }\\~~\vdots\\~n.~B\lor C}$$
I am sure you can fill out the rest of this &quot;proof by cases&quot;.