Should I perform a disjunctive syllogism directly on three expressions simultaneously? Consider
$$\begin{align*}
&1.\quad \lnot R \lor \lnot T \lor U\\
&2.\quad R\\
&3.\quad T
\end{align*}$$
It seems clear that you can end up with this:
$$4.\quad U$$
Now then, number $4$ was made by using disjunctive syllogism on three statements at the same time. Is that alright (for, say, a formal proof in a test), or am I supposed to do it "slower" and only do inferences with two statements at a time?
Another title for this question could have been: Is it alright to do inferences with more than two statements simultaneously? But, apparently, the destructive dilemma inference rule actually uses three statements anyway, so I guess I should keep this to just disjunctive syllogisms.
 A: $$\quad\quad \quad \;\;\;\;\;\;1. \quad \lnot R \lor \lnot T \lor U$$
$$2.\quad R$$
$$3. \quad T$$
$$4. \quad  U $$
Given the three premises $(1), (2), (3)$, yes indeed, $(4)$ is true. But assuming your task is to prove that $U$ follows from the first three premises: why wouldn't you add the intermediate step $(3.5)$, citing the premises and/or derivations used?:
$$(3.5)\; \lnot T \lor U\tag{ (1) (2) Disjunctive Syllogism}$$
$$(4)\; U \tag{ (3), (3.5) Disjunctive Syllogism}$$

You may want to speak to your instructor about how much detail to show. The disjunctive syllogism is typically taught as:
$$p \lor q$$
$$\lnot p$$
$$\therefore q$$
so you may also need to add $\lnot\lnot R$ from $(2)$, and $\lnot\lnot T$ from $(3)$, and then move to what I've numbered as $(3.5), (4)$, citing the respective premises to which double negation applies.
Again, it depends on your instructor, how explicit you need to be in your proofs.
A: Given:
$$1. ¬R∨¬T∨U$$
$$2. R$$ 
$$3. T$$ 
$$4. U$$ 
Using parentheses:
$$1. ¬R∨(¬T∨U)$$
Disjunctive syllogism is:
$$p∨q$$
$$¬p$$
$$∴q$$
Next step:
$$1. ¬R∨(¬T∨U)$$
$$2. R$$ 
$$3. T$$ 
$$4. U$$ 
$$5.  (¬T∨U) //justification: 1,2 DS $$
$$6.  U //justification: 3,5 DS $$
A: Here is the main question:

Is it alright to do inferences with more than two statements simultaneously?

The only inference rules one can formally use are those that have been formally defined.
Here is how the proof would look in a Fitch-style proof checker using disjunctive syllogism (DS) twice where I was forced to enter well-formed formulas and follow the inference rules:


Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/
