Everyone loves all lovers, Romeo loves Juliet $\vdash$ I love you (Natural deduction proof)
Yes, all the shortest proofs in natural deduction consist of 10 steps (included the 2 assumptions), because you have to apply 3 elimination rules for the quantifiers and 1 elimination rule for the implication, for 2 times.
For instance, if you translate the first assumption as $\forall x \forall y \forall z(Lxy \to Lzx)$ (which is logically equivalent to $\forall x (\exists y Lxy \to \forall y Lyx)$), the shortest proof in natural deduction is the following:
- $\forall x \forall y \forall z(Lxy \to Lzx)$ assumption
- $Lrj$ assumption
- $ \forall y \forall z(Lry \to Lzr)$ $ \ \forall_E$ from 1
- $ \forall z(Lrj \to Lzr)$ $\ \forall_E$ from 3
- $Lrj \to Lur$ $\ \forall_E$ from 4
- $Lur$ $\ \to_E$ from 5, 2
- $ \forall y \forall z(Luy \to Lzu)$ $\ \forall_E$ from 1
- $ \forall z(Lur \to Lzu)$ $\ \forall_E$ from 7
- $Lur \to Liu$ $\ \forall_E$ from 8
- $Liu$ $\ \to_E$ from 9, 6