From a comment, I have corrected my proof. Here's what I have now.
The Twin Prime Conjecture sates: There are infinitely many prime numbers $p$ for which $p+2$ is also a prime number.We consider 61 and 59, they are twin primes. $\phi(61)=60$ and $\sigma(59)=60$. So because $\phi(p)=p-1$ where p is prime and $\sigma(p)=p+1$ we can take that $m$ and $n$ are twin prime so $m=n+2$. So $m=n+2$ is equal to $m-1=n+1$ which is the same as $\phi(m)=\sigma(n)$, therefore $\phi(m)=\sigma(n)$ is ture for all twin prime integers. Hence if there are infinitely many twin primes then there are infinitely pairs of positive integers such that $\phi(m)=\sigma(n)$.