Manipulating Identities

• As a result of the substitution rule, the propositional variables appearing in the standard identities can be substituted (consistently) with arbitrary formulas and not just truth values.
• We can use this property to simplify a formula by rewriting it in a stepwise fashion.
• At each step, we use one of the standard identities (with appropriate substitutions) and thus conserve the truth table expressed by the formula.

 
• Example of Simplification
  Show P & (P v Q) P
• -P & ((R & S) v (R & -S)) & (P v Q)
  -P & Q & R
• -P & ((R & S) v (R & -S)) & (P v Q)
  -P & Q & R