could someone help me check if my proof is valid?
Use direct proof to prove the following theorem: $$ A \lor (B \rightarrow A), B \vdash_R A $$
We aren't allowed to use proof by resolution, we can only use logic axioms and inference rules such as hypothetical and disjunctive syllogism, constructive and destructive dilemma, modus ponens and modus tolens. Also, we can use similar equivalencies like contraposition $(A \Rightarrow B) \Leftrightarrow (\lnot B \Rightarrow \lnot A)$
Here is my proof:
- $A \lor(B \rightarrow A)$, premise
- $B$, premise
- $\lnot A$, (assumption)
- $\lnot A \rightarrow(B \rightarrow A)$, elimination of disjunction from (1)
- $B\rightarrow A $, (modus ponens from (3), (4))
- $A$, (modus ponens from (2), (5))
The reason I'm asking is because I'm not sure if it is valid, since I made an assumption that A is incorrect and used it as a premise until I got to a contradiction at $6.$
Since I have used an incorrect assumption as a premise, should I start anew but using the assumption that A is true, albeit me getting a contradiction?