Deductive Logic Proof I'm struggling with a homework problem here and I'm trying to figure out how to proceed. The problem is the following:


*

*Determine if the following argument is valid or not, given the premises 1-3. If the argument is valid, give deductive proof. If it is not, use the truth table method to point out the flaw.


*

*if the engine works, then the control light is on, provided that the battery is not dead.

*if the battery is dead, then the engine does not work.

*if the control light is on, then the engine works. 
Conclusion: if the battery is not dead, then the engine works, and the control light is on.
I understand how to navigate these types of problems and can almost see the answer myself, I'm just not quite sure how to deduce it in proper logic. I was using the following...
B = battery is on
C = control light is on
E = engine works
Any help would be appreciated. Thanks.
 A: We have the following premises and conclusion:


*

*If the engine works, then the control light is on, provided that the battery is not dead.

*If the battery is dead, then the engine does not work.

*If the control light is on, then the engine works.

Conclusion: If the battery is not dead, then the engine works, and the control light is on.

We have the following symbolization key:

B = battery is on
C = control light is on
E = engine works

Given the above we can rewrite the premises and conclusion as follows:

*

*$(E ∧ B) → C$

*$¬B → ¬E$

*$C → E$

*$∴ B → (E ∧ C)$
We need to provide a deductive proof for the following or show using the truth table method that there is a flaw in the argument.
$$(E ∧ B) → C, ¬B → ¬E, C → E ∴ B → (E ∧ C)$$
The conclusion that the battery is on is sufficient for both the engine to work and the control light to be on seems suspect. So we can try the truth table method to see if the argument represents a tautology or not.
Here is the truth table:

The fourth line in the truth table shows where the argument as symbolized has true premises but a false conclusion and so it is not valid.
A counterexample would be when $B$ is true but $C$ and $E$ are false.

Michael Rieppel. Truth Table Generator. https://mrieppel.net/prog/truthtable.html
