Let P, Q, R and S be Propositions. Assume that the equivalences P ⇔ (Q ∨ ¬ Q)…

Key observations: Q ∨ ¬Q is a tautology, so P ⇔ (Q ∨ ¬Q) makes P equivalent to True. Also Q ⇔ R means R has the same truth value as Q.

• Replace P by True:

  (P ∧ Q) ⇒ ((P ∧ R) ∨ S) becomes (True ∧ Q) ⇒ ((True ∧ R) ∨ S).

• Simplify the conjunctions:

  (True ∧ Q) is Q and (True ∧ R) is R, so the formula is Q ⇒ (R ∨ S).

• Use Q ⇔ R to replace R by Q:

  This yields Q ⇒ (Q ∨ S).

• Final simplification:

  Q ⇒ (Q ∨ S) is always true because whenever Q is true, Q ∨ S is certainly true.

Conclusion: Therefore the given formula is always true.