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

Apply [L1], [L2], and [L3] in order to obtain an equivalent grammar $G_1$ in which every remaining production is nonunit and every $\epsilon$-production is removed except possibly a start-symbol rule.

For each terminal $a$ that appears in a right-hand side of length at least $2$, introduce a fresh variable $T_a$ with production $T_a\to a$, and replace that occurrence of $a$ by $T_a$. This changes no generated word, because each use of $T_a$ can immediately be expanded to the terminal $a$. After this step, every right-hand side is either a single terminal or a word of variables only.

For each production $A\to B_1{B}_2\cdots B_k$ with $k\ge 3$, introduce fresh variables $C_1,\dots ,C_{k-2}$ and replace the production by $A\to B_1{C}_1,C_i\to B_{i+1}{C}_{i+1}(1\le i\le k-3),C_{k-2}\to B_{k-1}{B}_k\mathrm{.}$ The middle family is empty when $k=3$, so that case is the two-rule chain $A\to B_1{C}_1$ and $C_1\to B_2{B}_3$. One use of the original long rule is equivalent to this forced chain of binary rules, and conversely the binary chain produces exactly the same string of variables.

After steps 2.1 and 3.1, every production is either $A\to BC$, $A\to a$, or the exceptional start-symbol rule $S\to \epsilon$. Therefore [L4] says the resulting grammar is in Chomsky normal form, and the preceding steps preserved the language.