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

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To find logical statements equivalent to (A ∧ B) ↔ (A ∧ C) without using truth tables, users can refer to established logical theorems and argument forms. Substituting equivalent statements, such as D = (A ∧ B), can help derive new equivalents. A helpful resource mentioned is a Wikipedia table of valid argument forms in propositional logic. One proposed equivalent is A ∧ (B ↔ C). Exploring these resources can simplify the process of finding logical equivalences.
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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.
 
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That helps. Thanks.
 
I am pretty sure that this is equivalent:
( A \wedge ( B \leftrightarrow C ) )
 

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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