Help me out please!! I have been trying to solve it for hours
closed as off-topic by Mauro ALLEGRANZA, Eliran, Not_Here, christo183, virmaior Dec 1 '18 at 5:53
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You wish to prove a negation statement, so ... use a Proof of Negation.
Accept the premises, assume the positive, derive a contradiction, therefore deduce that the negation is derivable from the premises.
| A → ¬B |_ ¬A → ¬C | |_ B ∧ C | | : | | : | | ┴ | ¬(B ∧ C)
P1: A → ¬B
P2: ¬A → ¬C
Always true: (A ∨ ¬A)
¬B ∨ ¬C
This will be a partial answer.
Note that the premises are two conditionals. One of them has A and the other has ¬A as the antecedent of the conditional. This suggests that the law of the excluded middle might be useful in deriving the goal.
Also note that the goal can be transformed using De Morgan rules from ¬(B∧C) to ¬B ∨ ¬C. Since that is a disjunction if one can get ¬B one can use disjunction introduction to get ¬B ∨ ¬C. This would work also if one could derive ¬C.
This may be all you need as a hint to prove this.
It is sometimes useful to have supplementary texts to help clarify answers. You might try the textbook forallx and the proof checker linked to in the references.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/