Prove:

        (d)    ,

with no assumptions.

We prove (d) by the deduction rule.

We assume

        (1)   

and show

        (2)    .

Formula (2) is transformed into:

        (3)    .

Formula (3) is transformed into:

        (4)    .

We prove (4) by proving the first alternative negating the other(s).

We assume

        (6)    .

We now show

        (5)    .

From (6) and (1) we obtain by modus tollens

        (7)    .

Formula (7) is simplified to:

        (8)    .

Formula (5) is true because it is identical to (8).

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Converted by Mathematica      February 11, 2000