Policy ineffectiveness proposition

Suppose,

the Aggregate supply is given by the Lucas Supply Curve –

$y_t = ȳ + b(P_t – E_{t-1}P_t) + μ_t$

where $μ_t$ is stochastic supply shock (following standard normal error properties).

Aggregate demand equation is given by –

$y_t = m_t – P_t + v_t$

where $v_t$ is stochastic demand shock (following standard normal error properties).

Monetary authorities follows the policy rule –

$m_t = bar{m} + m_{t-1} – cy_{t-1} + dμ_t +fv_{t-1}$

For the system, after solving for $y_t$ under the assumption of rational expectations (ie. Agents incorporate monetary policy changes into their decisions), I get it as a function of demand and supply shock.

$y_t$ = $b/(1+b) v_t$ + $(1+bd)/(1+b) μ_t$

Here, though $y_t$ is a function of policy parameter ‘d’ but $μ_t$, being supply shock of the current period equally random to both public and monetary authorities, is unanticipated part of money supply.

PIP argues that any anticipated changes in money supply cannot affect real variables. Since, $μ_t$ is unanticipated so by this regard PIP must hold. But since the policy parameter ‘d’ enters the output decision so monetary policy do have some influence over real variable. Then does it mean PIP doesn’t hold.

I am confused between the two arguments. It will be helpful if someone can explain which one of the two is right? (Whether PIP holds or not and why?)