Tautology and contradiction Theorem Let A and B be compound statements. If the compound statement A and A = ⇒ B are tautologies, then so is the compound statement B . The substitution theorem for tautologies Let A be a tautology and suppose that A contains the distinct statement variables p 1 , p 2 , ..., p n (and perhaps others as well). Suppose that B 1 ,B 2 , ...,B n are compound statements. Then, if in the tautology A we replace p 1 by B 1 , p 2 by B 2 , and so on, the resulting statement is also a tautology. Example p ∨ ( ∼ p ) is a tautology. If we replace p by p = ⇒ ( ∼ q ) , then the statement [ p = ⇒ ( ∼ q )] ∨ ∼ [ p = ⇒ ( ∼ q )] must be a tautology.
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