How to find a logical statement equivalent to the one I have?

[omoplata]

I have the following logical statement
$$(A \wedge B) \Longleftrightarrow (A \wedge C)$$
I want to find other logical statements that are equivalent to this. But I don't want to draw truth tables.

Is there a list of 'theorems' that I can look up somewhere? For example, if I have an equivalent statement for $D \Longleftrightarrow (A \wedge C)$, then I can maybe substitute $D = (A \wedge B)$ and maybe come up with something.

Sorry if my terminology is wrong. Logic is not exactly my subject.

 

[JThompson]

The closest thing I could find is a Wikipedia table of valid argument forms in propositional logic.

http://en.wikipedia.org/wiki/Propositional_logic#Basic_and_derived_argument_forms"

 

[omoplata]

That helps. Thanks.

 

[praeclarum]

I am pretty sure that this is equivalent:
( A $\wedge$ ( B $\leftrightarrow$ C ) )

 

[blue_raver22]

There are several ways to find logical statements that are equivalent to the one you have. One approach is to use logical equivalences, which are rules that allow you to manipulate logical statements while preserving their truth value. These rules can be found in any introductory logic textbook or online resource.

For example, one logical equivalence that is relevant to your statement is the distributive law, which states that (A \wedge B) \Longleftrightarrow (A \wedge C) is equivalent to A \wedge (B \Longleftrightarrow C). So, you could substitute D = (B \Longleftrightarrow C) and get D \Longleftrightarrow A \wedge D, which is equivalent to your original statement.

Another approach is to use logical inference rules, such as modus ponens or modus tollens, to derive equivalent statements. These rules allow you to make deductions based on the logical structure of a statement. For example, if you have the statement A \rightarrow B and you know that A is true, you can infer that B is also true. Using these rules, you can manipulate your original statement to derive equivalent statements without having to draw truth tables.

In terms of finding a list of theorems, there are many resources available online that provide lists of logical equivalences and inference rules. Some examples include the Stanford Encyclopedia of Philosophy, the Internet Encyclopedia of Philosophy, and various logic textbooks. It may also be helpful to consult with a logic expert or take a logic course to gain a deeper understanding of logical equivalences and how to use them effectively.