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Help me out please!! I have been trying to solve it for hours

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    This site has been recently "submerged" by "homework question": this is not the original aim of the site. – Mauro ALLEGRANZA Nov 29 '18 at 15:57
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    If you want help with a homework question you need to show at the very minimum that you have an understanding of what is being asked and that you've attempted to solve it yourself. – Not_Here Nov 29 '18 at 16:48
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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)
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P1: A → ¬B

P2: ¬A → ¬C

Always true: (A ∨ ¬A)

————

¬B ∨ ¬C

————

¬(B∧C)

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    This answer is amazingly spartan on explanation. Care to help the OP understand the ideas here? (this is correct after a sort and sufficiently clear for people who already know the answer). – virmaior Dec 1 '18 at 5:52
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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.


References

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

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/

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    Not grasping at all why this was downvoted. – virmaior Dec 1 '18 at 5:51

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