For all integers $n$: if $7n+4$ is even, then $5n+6$ is even. I am still new to the proof game so please be kind! This is my third time attempting a proof. Any feedback would be greatly appreciated. Thank you in advance.
Claim: For all integers $n$: if $7n+4$ is even, then $5n+6$ is even.
Proof: Assume $5n+6$ is odd. If $5n+6$ is odd, then $5n+6=2k+7 \Rightarrow 5n=2k+1$ for some $n,k \in \mathbb{Z}$. If $7n+4$ is even, then $7n+4=2k$ for some $k \in \mathbb{Z}$. Therefore, $5n=7n+4+1 \Rightarrow 2n=-5$ for some $n \in \mathbb{Z}$. Clearly, $2n \neq -5$, which contradicts our assumption that $5n+6$ is odd.
I feel like I completely lost myself, but do not know where to go. Please be kind! Any words of wisdom and insight would be great. Thanks.

Modulo $2$ we have by assumption
$$
0\equiv 7n+4\equiv 1\cdot n+0=n,
$$
so that
$$
5n+6\equiv 5\cdot 0+0\equiv 0.
$$
