What does a proof that Co-NP =P entail for the NP versus Co-NP question
$\text{P}=\text{co-NP}$ implies that $\text{co-P}=\text{co-(co-NP)}=\text{NP}$.  But 
$\text{co-P}=\text{P}$: you can just swap the accept and reject states of a deterministic Turing machine to complement the language that it decides.