# Applying rules of inference in natural deduction

When applying a rule of inference, is it okay to "skip" a step (i.e, apply a same rule to multiple parts of a statement)?

For example:

1. (A > B) ^ (C > D)

2. (~A v B) ^ (~C v D) 1, Impl.

As opposed to applying simplification first, then implication on each statement alone (resulting in more lines).

It seems like more of a stylistic point, but I'm curious and would like to keep to best practices.

## 1 Answer

In general, I would say YES, unless the problem explicitly asks for a detailed step-by-step proof "mechanically" testable.

But you have to take care of what rules are you using.

Consider the set of rules referred into the post : step-by-step natural deduction.

As you can see, there are two types of rules :

those, called "conditionally true", that cannot be applied to a sub-formula inside a formula;

and :

those that are "biconditional rules" (called : rules of equivalence), that can be applied substituting a sub-formula inside a formula.

• Thank you for your help! I asked the previous question too and your answer was very insightful. – person May 23 '18 at 15:07
• So suppose we had a premise like: (A > B) ^ (A) , could we simply move immediately to the next line and say B, referring only to that line and to Modus Ponens? Or would that one need to be simplified as well? I wish I had a resource for these conventions – person May 23 '18 at 15:09
• Got it, thanks! And I forgot my last password sadly – person May 23 '18 at 18:12