Consider the following argument with premise $\forall x(P(x)\vee Q(x))$ and conclusion $(\forall xP(x))\wedge(\forall xQ(x))$
| (A) $\forall x(P(x)\vee Q(x))$ | Premise |
| (B) $P(c)\vee Q(c)$ | Universal instantiation from (A) |
| (C) $P(c)$ | Simplification from (B) |
| (D) $\forall xP(x)$ | Universal generalization of (C) |
| (E) $Q(c)$ | Simplification from (B) |
| (F) $\forall xQ(x)$ | Universal generalization of (E) |
| (G) $(\forall xP(x))\wedge(\forall xQ(x))$ | Conjunction of (D) and (F) |
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