Every context-free language has an equivalent grammar in Greibach normal form

Replace $G$ by an equivalent CNF grammar $G_1$ as in [L1], and order its original variables as $A_1,\dots ,A_n$. In a forward pass through this list, fix $A_i$ and repeatedly use [L2] to remove every production of the form $A_i\to A_j\alpha$ with $j<i$. When that is done, use [L3] to eliminate the remaining immediate left-recursive productions $A_i\to A_i\beta$, introducing an auxiliary variable $B_i$ if needed. After stage $i$, every production of $A_i$ begins either with a terminal or with an original variable $A_k$ satisfying $k>i$, and every new production of $B_i$ begins either with a terminal or with an original variable.

In particular, after the forward pass the productions of $A_n$ already begin with terminals, because there is no index $k>n$. Now argue downward on $i=n-1,n-2,\dots ,1$. Assume the productions of $A_{i+1},\dots ,A_n$ already begin with terminals. Every production of $A_i$ begins either with a terminal or with some $A_k$ having $k>i$ by step 1.1, so repeated use of [L2] substitutes the full terminal-leading production family of each such $A_k$ into the leading occurrence. Thus every production of $A_i$ can also be made terminal-leading without changing the language.

The same downward substitutions clean every auxiliary variable introduced during the left-recursion eliminations. By step 1.1, each production of such a $B_i$ already begins either with a terminal or with an original variable $A_k$. Once the productions of each $A_k$ are terminal-leading by step 2.1, repeated use of [L2] removes that leading variable from the productions of $B_i$ as well. Therefore, after finitely many substitutions, every production in the whole grammar begins with a terminal, except possibly the start-symbol rule $S\to \epsilon$.

Because the construction started from CNF, the symbols after the first terminal in any rewritten production are variables, and the optional start-symbol rule $S\to \epsilon$ is preserved only in the exceptional case allowed by [L4]. Hence the final grammar is in Greibach normal form, and [L2] and [L3] preserved equivalence throughout.