Edges not contained in triangles and the number of contractible edges in a 4-connected graph

Let GG be a 4-connected graph, and let Ec(G)Ec(G) denote the set of 4-contractible edges of GG and let Ẽ(G) denote the set of those edges of GG which are not contained in a triangle. Under this notation, we show that if |Ẽ(G)|≥15, then we have |Ec(G)|≥(|Ẽ(G)|+8)/4.