Note: I'm going to use "<->" for "if and only if" and "*" for "and"
1. B <-> W
2. L -> B
3. ~B v ~W :. ~(W v L)
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4. (B -> W) * (W -> B) 1 Material Equivalence
5. (W -> B) * (B -> W) 4 Commutation
6. B -> W 4 Simplification
7. W -> B 5 Simplification
8. B -> ~W 3 Material Implication
9. W -> ~W 7,8 Hypothetical Syllogism
10. ~W v ~W 9 Material Implication
11. ~W 10 Tautology
12. ~B 6,11 Modus Tollens
13. ~L 2,12 Modus Tollens
14. ~W * ~L 11,13 Conjunction
15. ~(W v L) 14 De Morgan's Law