I am trying to use proof of sequence correctly to make valid

Question

Here I am trying to use a proof sequence so that the argument is valid (hint: the last A’ has to be inferred).

(A → C) ∧ (C → B') ∧ B → A'

Here are my steps I tried but not sure if this is correct

1. A → C = Conjunctive Simplification
2. C → B' = Conjunctive Simplification
3. A → B' = Chain Rule
4. B = Conjunctive Simplification
5. A' = Contrapositive

Answer 1

Your steps would be valid if you are trying to prove: $(A\to C)\wedge (C\to B')\wedge B\;\to\; A'$

$$\begin{array}{l|lrc} 0 & (A\to C)\wedge (C\to B')\wedge B & \text{Assumption} \\[1ex] 1 & (A \to C) & 0, \text{ Conjunctive Simplification} & P\wedge Q\vdash P \\ && \text{aka Conjunction Elimination} \\[1ex] 2 & (C \to B') & 0, \text{ Conjunctive Simplification} \\[1ex] 3 & (A \to B') & 1,2, \text{ Hypothetical Syllogism} & P\to Q, Q\to R\vdash P\to R \\ && \text{aka the Chain Argument} \\[1ex] 4 & B & 0, \text{ Conjunctive Simplification} \\[1ex] 4a & B'' & 4, \text{ Double Negation} & P\vdash P'' \\[1ex] 5 & A' & 3, 4a, \text{ Modus Tollens} & P\to Q, Q'\vdash P' \\[1ex] \hline \therefore & \Big((A\to C)\wedge (C\to B')\wedge B\Big)\;\to\; A' & \Box \end{array}$$