For equal-sized matrices, row equivalence, equality of row spaces and equality of reduced row echelon forms are equivalent

If $A$ and $B$ are row equivalent, [L2] gives equal row spaces, and reducing either one gives a reduced form in the same row-equivalence class, so [L1] gives the same RREF.

Suppose the row spaces are the same space $W$. If the nonzero rows of an RREF have pivots $p_1<\cdots<p_r$, then [L3] expresses every nonzero $w\in W$ uniquely as a linear combination of those rows. The first row with nonzero coefficient makes the leading position of $w$ equal to its pivot. Hence the pivot columns are exactly the possible leading positions of nonzero vectors in $W$, so both RREFs have the same pivots. Corresponding nonzero rows have $1$ in the same pivot and $0$ in every other pivot column. Their difference lies in $W$ and vanishes at every possible leading position, so it is zero. The nonzero rows, and then the zero rows, agree.

If $A$ and $B$ have a common RREF, reverse the reduction from $B$ to that form using [L4] and concatenate it with the reduction from $A$; this gives a row reduction from $A$ to $B$.