Let $k$ be an algebraically closed field, and consider some subscheme $X\subset \mathbb{P}_k^n$. Let $x$ be a closed point of $X$, and $H$ a general hyperplane containing $x$. There is a regular map $\phi:\mathbb{P}^n\setminus{\{x\}}\rightarrow \mathbb{P}^{n-1}$ gotten by projecting from the point $x$.

My question: Is $\overline{\phi(H\setminus x)}\cap\overline{\phi(X\setminus x)}=\overline{\phi((X\cap H)\setminus x)}$?

This can be rephrased in terms of commutative algebra: Let $A=k[x_1,\ldots,x_n]$, $B=A[x_0]$, $I$ be a homogeneous ideal of $B$, and $h$ a general linear form of $A$. The question above is equivalent to the following: Is $(I\cap A)+h=(I+h)\cap A$?

More generally, what if we replace $H$ and $h$ by a general hypersurface/general polynomial of degree $d$?

If $h$ or $H$ isn't general, equality doesn't hold, even for $d=1$. Consider for example $n=3$ and take $I=\langle x_0x_1,x_2^2+x_0x_3\rangle$, and consider $h=x_3-x_1$. Then $I\cap A=\langle x_1x_2^2 \rangle$, so $(I\cap A)+h=\langle x_1x_2^2,x_1-x_3\rangle$, but $(I+h)\cap A=\langle x_2^2,x_1-x_3\rangle$.