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Two rank-$d$ complex vector bundles over ${\mathbb{CP}}^d$ are isomorphic exactly when their Chern classes agree. Equality of Chern classes gives equality of their Chern characters; the Chern character is injective here because $K^0({\mathbb{CP}}^d)$ is torsion-free. The bundles are therefore stably isomorphic, and stable cancellation for complex vector bundles applies because ${\mathbb{CP}}^d$ has real dimension $2d$.

For a compact space $X$, $K^0(X)$ is the Grothendieck group of isomorphism classes of finite-rank complex vector bundles under direct sum. Tensor product descends to this group and makes it a ring, with $[{\mathbb{C}}_X]$ as its unit.

The hypothesis $E_0\oplus {\mathbb{C}}_X\cong E_1\oplus {\mathbb{C}}_X$ says that $E_0$ and $E_1$ define the same stable class. Rank-$d$ bundles are classified by maps to $BU(d)$, and the stabilization $BU(d)\to BU$ is $2d$-connected. Since the finite CW complex $X$ has dimension $k\le 2d$, stabilization is injective on $[X,BU(d)]$. Thus stable cancellation for complex vector bundles gives

Now let $E_0,E_1$ have rank $d$ over Complex projective space ${\mathbb{CP}}^d$ and have the same Chern classes. Their Chern characters agree because each component of $\mathrm{ch}(E)$ is a universal rational polynomial in the Chern classes. The ringis torsion-free, while the Chern character becomes an isomorphism after tensoring with $\mathbb{Q}$; it is therefore injective. Hence $[E_0]=[E_1]$ in K-theory, so after adding trivial bundles they are isomorphic. Since ${\mathbb{CP}}^d$ has real dimension $2d$, cancellation applies once more: