Prove, that for any A, B, C sets, $A\times (B\setminus C) = (A\times B)\setminus (A\times C)$ is true.

Question

I have to prove the equation $$A\times (B\setminus C) = (A\times B)\setminus (A\times C)$$ knowing that $$A\times B = \{<a,b> |\: a\in A \wedge b\in B\}$$ which I attempted to do the following way:
Let's take any x.
Then $x\in A\times (B\setminus C)$ iff $$(\exists a\in A)\: (\exists b\in (B\setminus C))\: (x=<a,b>)$$ $$\equiv \exists a\exists b \:(a\in A) \wedge (b \in B \:\wedge b \notin C) \:\wedge x=<a,b>$$ $$\equiv \exists a\exists b \: (a \in A\: \wedge b \in B\: \wedge x=<a,b>) \: \wedge (a\in A\: \wedge b\notin C\: \wedge x=<a,b>)$$ Further transformations lead me to $(A\times B) \times(A\setminus C)$, which is not the thing I was meant to prove. I need an advice how should I do it the right way, because I cannot find any solution.

Answer 1

What you should have written is $$(a,\,b)\in A\times (B\backslash C)\iff a\in A\land b\in B\land b\not\in C\\\iff (a\in A\land b\in B)\land\neg(a\in A\land b\in C)\\\iff (a,\,b)\in (A\times B)\backslash (A\times C).$$