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If every monomorphism is strong, let $m:A\to B$ be both a monomorphism and an epimorphism. Use $m$ as both vertical maps in the lifting square, with horizontal maps $1_A$ and $1_B$. The strong monomorphism property gives $d:B\to A$ satisfying $dm=1_A$ and $md=1_B$. Thus $m$ is an isomorphism, proving that the category is balanced.

For the converse, take $e:X\to Y$ epic, $m:A\to B$ monic, and $mu=ve$. Form the pullback in a category of $m$ along $v$:The equality $mu=ve$ gives $t:X\to P$ with $pt=e$ and $qt=u$. By preservation of monomorphisms under pullback in a category, $p$ is monic. It is also epic: if $hp=kp$, then $he=hpt=kpt=ke$, and cancellation of $e$ gives $h=k$. The balanced category hypothesis therefore makes $p$ an isomorphism.

Set $d=q{p}^{-1}$. Then $md=v$ and $de=qt=u$. Any other such lift agrees with $d$ after composing with the monomorphism $m$, so it is equal to $d$. Consequently in a category with pullbacks in a category, being balanced is equivalent to every monomorphism being a strong monomorphism. This is the balanced categories with pullbacks have strong monomorphisms principle; the proof does not assume pullback stability of epimorphisms.