# Simple Fitch Proof without using DeMorgan's Law

Can someone tell me how to prove B → ¬A given the premises 1: (B ∧ A) → D and 2: (B ∧ A) → D using the Fitch system?

I have been trying to solve this proof using DeMorgan's law, but I am unable to as this proof is bound by Fitch rules (= intro, = elim, ^ into, ^ elim, etc.) -- I'll link the Fitch Rule Summary below:

https://www.ocf.berkeley.edu/~brianwc/courses/logic/rulesummary.html

I was trying to solve this proof using DeMorgan's law, Double Negation Law, Simplification Law, and MT law, but once again, I am bound by Fitch rules.

These are the premises and goal:

1: ¬(D ∨ ¬C)

2: (B ∧ A) → D

Goal: B → ¬A

Without DeMorgan law it's usually not that straightforward and you need more design for your proof search as following sketched:

B  ---assume
A  ---assume
B ∧ A  ---∧ intro
(B ∧ A) → D  ---premise
D  --- → elim (MP)
D ∨ ¬C ---∨ intro
¬(D ∨ ¬C)  ---premise
⊥  ---⊥ intro
¬A  --- ¬ intro
B → ¬A --- → intro

• Yeah-- that's the struggle I'm facing as well. With DeMorgan's law I was able to mostly solve the proof, but utilizing Fitch rules I have been struggling. I was hoping someone more well-versed with Fitch could be of aid, but it's also not as commonly used as DeMorgan's would be. Thanks for the advice though, I appreciate it! Commented Dec 15, 2021 at 0:43