If ax+by+cz = d has an integer solution, then ax+by = d has an integer solution.
Disproof: A counterexample is $ a = b = c = 1$ and $d = 3$. Then $(x, y, z) = (1, 1, 1)$ is a solution to $ax + by + cz = d$ but $(x, y) = (1, 1)$ is not an integer solution to $ax + by = d$.
There is also another problem about it. For illustration, the equation $2x+3y+4z=1$ has integer solution because $\gcd(2,3,4)=1$, while the equation $2x+4z=1$ fails to be solved. On the other hand the equation $2x+3y+5z=1$ is solvable and also each of equations $2x+3y=1$, $2x+5z=1$ and $3x+5z=1$.
To summarize, your verification is true and is false!!!