replication - Replicating strategy in the Black-Scholes model

I have a two-asset Black-Scholes model for a financial market:

$dB_t=B_t r dt$

$dS_t=S_t(\mu dt+\sigma dW_t)$

I introduce a European claim $\xi=max(K,S_T)$ with maturity $T$, for some fixed $K$. I have calculated what the no-arbitrage price of this claim should be at each time $t

I know that if $V(t,S)$ is a solution to the Black-Scholes PDE subject to the terminal condition $V(T,S)=\max(K,S)$, then $V(t,St)$ is a no-arbitrage time-$t$ price for $\xi$, and that the trading strategy given by taking initial wealth to be $V(0,S_0)$ and the time-$t$ holding in the stock to be $\frac{\partial V}{\partial S}$ is a replicating strategy for the claim.

If I view the pricing function I originally found (by computing expectations) as a function $\xi(t,St)$, is it necessarily true taking initial wealth to be $\xi(0,S_0)$ and taking time-t holding in the stock to be $\frac{\partial \xi}{\partial S}$ will give a replicating portfolio? It should be, simply due to the fact there is a unique equivalent martingale measure in this market, so there must be a unique no-arbitrage time -$t$ cost for the claim at each time $t$, and so $\xi(t,S_t)$ must solve the Black-Scholes PDE.

My question is, is it possible to prove that this trading strategy does replicate the claim without appealing to the fact that the pricing function solves the Black-Scholes PDE?