The first definition of antisymmetric of a relation is $(a,b) \in R$ and $(b,a) \in R \implies a=b$. There is another saying if $a \neq b$ and $(a,b)\in R$, then $(b,a) \not\in R$. I could intuitively understand this or even by truth table, but I want to show how to transform one into another. I had gone with contrapositive and logical equivalence, neither work out. Here is what I got so far: $a\neq b \implies (a,b)\not\in R $ or $(b,a)\not\in R$ I cannot go further with this one.
Could someone show me how to do it, please?
Let $p$ be $(a,b)\in R$, $q$ be $(b,a)\in R$, and $r$ be $a=b$.
$(p\wedge q)\to r$ is equivalent to $p\to( q\to r)$ is equivalent to $p\to( \neg r\to \neg q)$ is equivalent to $(p\wedge \neg r)\to \neg q$.