“cooperating provably results in a better payoff than defecting”. This is enough to convince the agent to cooperate. But if you switch C and D, you end up with a proposition that means “cooperating provably results in a worse payoff than defecting”. This is not enough to convince the agent to defect. The agent will defect if there is no proof that cooperating results in a better payoff than defecting. So at least one would have to say a bit more here in the proof.
One can try to take the proof of Proposition 3 and switch C and D around, but one quickly runs into difficulty: The second line of that proof contains
“cooperating provably results in a better payoff than defecting”. This is enough to convince the agent to cooperate. But if you switch C and D, you end up with a proposition that means “cooperating provably results in a worse payoff than defecting”. This is not enough to convince the agent to defect. The agent will defect if there is no proof that cooperating results in a better payoff than defecting. So at least one would have to say a bit more here in the proof.