Goldbach Conjecture implies Weak Goldbach Conjecture

Theorem
The Goldbach Conjecture:
 * Every even integer greater than $2$ is the sum of two primes

implies Goldbach's Weak Conjecture:
 * Every odd integer greater than $7$ is the sum of three odd primes.

Proof
Take any odd integer $n$ such that $n > 7$.

Then $m = n - 3$ is an even integer $n$ such that $m > 4$.

If the Goldbach Conjecture holds, then $m$ is the sum of two primes: $m = p_1 + p_2$.

If one of them were $2$, then $m - 2$ would have to be even, which if it is prime it can not be.

So if $m > 4$, both $p_1$ and $p_2$ must be odd.

So then we have that $n = p_1 + p_2 + 3$, that is, the sum of three odd primes.

So, if the Goldbach Conjecture holds, then so does Goldbach's Weak Conjecture.