In 1964 Vizing proved that starting from any k-edge-coloring of a graph G one can reach, using only Kempe swaps, a ($Δ$ + 1)-edge-coloring of G where $Δ$ is the maximum degree of G. One year later he conjectured that one can also reach a $Δ$-edge-coloring of G if there exists one. Bonamy et. al proved that the conjecture is true for the case of triangle-free graphs. In this paper we prove the conjecture for all graphs.