I'm taking a lecture in graphtheory at the moment and I'm having trouble to understand the proof for the following theorem by C. Thomassen (1980) since the one given in the lecture seems incomplete.
$$\text{Every } 3\text{-connected graph } G=(V,E) \text{ with } |V| \geq 5 \text{ has one } xy \in E \text{ so that }\\ G/xy \text{ is still } 3\text{-connected}$$
Where $$G/xy = ((V \setminus \{x,y\}) \;\dot\cup \;(z_e),\; E')$$ with $$E' = \{zz_e \mid z \not \in \{x,y\}, zx \in E \lor zy \in E\} \cup \{zz' \in E \mid \{z,z'\} \cap \{x,y\} \not = \emptyset\}$$
I've tried to search here and via google but couldn't find the proof for this. Some additional resource containing the proof would be great!