Direct proof for $a \rightarrow b, c \rightarrow b, d \rightarrow (a \lor c), d \Rightarrow b$

Question

I am just starting learning proofs in my discrete math class. I need to find the direct proof for $a \rightarrow b, c \rightarrow b, d \rightarrow (a \lor c), d \Rightarrow b$.

These are my steps:

1. $ a \rightarrow b [Premise]$
2. $ c \rightarrow b [Premise]$
3. $ (a \rightarrow b) \land (c \rightarrow b) [Conjunction 1, 2] $
4. $ d \rightarrow ( a \lor c) [Premise]$
5. $d \rightarrow b [Constructive Dilemma 3, 4]\square$

Is this a correct proof? I tried multiple ways and nothing worked, but I am not sure if I am using Constructive Dilemma the right way.

Answer 1

Hint

Let assume that $⇒$ means $\vdash$, i.e. consequence.

Use Modus Ponens with $d$ and $d \to (a \lor c)$ to derive : $a \lor c$.

Then use Disjunction elimination with $a \lor c$ and the first two premises to derive the conclusion $b$.