I'm working on writing a proof and I'm a bit stuck on one part in particular.

Here's an example of the type of problem I'm working with.

$Let\ C(x)\ and\ P(x)\ be\ defined\ as:$

$C(x) = "This\ post\ was\ uploaded\ on\ x."$ $P(x) ="x\ comes\ after\ Monday\ and\ before\ Wednesday"$.

$\ \ \ \ \ \$ If I were to write: $C("Tuesday") \neq P("Tuesday")$ the answer would be $False$ (today is Tuesday), however I want to compare the contents of the two statements, not the logical outcome. I want to show that the two statements $P(x)$ and $C(x)$ are different because they are talking about different things -- getting a $True$ value (in this case) as output. Specific context within the statements isn't really important to the larger proof I'm working on; the specific contents don't matter, just the fact that they're different. The order in which things were stated doesn't matter either.

How would one go about writing this using discrete notation?

• Thanks for the edit. Using MathJax would allow $\lnot \forall x\; (C(x) \iff P(x))$. Commented Jan 4, 2018 at 15:10