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(A intersection B)' = A' U B'

I know that it is proof is along the lines of

Let x be in (A intersection B)'.

Then x is NOT in A intersection B

<=> x not in A AND x not in B (use logical connectors, negations, etc)

<=> x in A' OR x in B'

<=> x in A' U B'

but my question is how does the AND become an OR?