For a given maturity, given three equally spaced option strikes $K_1,K_2,K_3$ a "butterfly" combination consists of shorting 2 of the middle strike calls and buying one each of the "wing" or lateral calls. This position has a positive cost i.e. $c_1+c_3-2 c_2 >=0$ (why? because it has a positive payoff for $S_T\approx K_2$ and zero payoff elsewhere).
Into the first equation we can substitute $C$ and $P$ as given by the other two equations, we get:
$(S-K)^+ -(K-S)^+ +TV_C - TV_P = S-PV(K)$
If interest rates are zero then $PV(K)=K$ and then we indeed have
Note: as suggested in the comments above a slightly different definition of Intrinsic Value ...