I think there is a mistake in your equation : (A-B)∪(A-C) ≠ (A∩B)∪(A∩C).
Here it's a general example to verify it:
A={a,b,c,d} , B={a,c} and C={c,d}
A-(B∩C) = {a,b,c,d} - ({a,c}∩{c,d}} = {a,b,c,d} - {c} = {a,b,d}
And (A-B)∪(A-C) = {b,d}∪{a,b} = {a,b,d}
Then : A-(B∩C) = (A-B)∪(A-C).
But...