How to get from $P [\phi\leftrightarrow \psi]=1$ to $P[\phi]=P[\psi]$

Question

That's the formulas i got

$\begin{align}P[\phi \leftrightarrow \psi] = 1 &\equiv P[(\phi\wedge\psi)\vee(\neg\phi\wedge\neg\psi)] = 1 \\ &\equiv P[\phi\wedge\psi] + P[\neg\phi\wedge \neg\psi] = 1 \\ &\equiv P[\phi\wedge\psi] = 1−P[\neg\phi\wedge\neg\psi] \\ &\equiv P[\phi\wedge\psi] = P[\phi\vee\psi] \\ &\equiv P[\phi\wedge\psi] = P[\phi] + P[\psi]−P[\phi\wedge\psi]\end{align}$

Need some advice on how to proceed

Answer 1

Notice that $P[\phi] = P[\phi \land \lnot\psi] + P[\phi \land \psi] = P[\phi \land \psi]$.

This lets you, from

$\begin{align} P[\phi \land \psi] & = P[\phi] + P[\psi] - P[\phi \land \psi]\\ & = P[\phi \land \psi] + P[\psi] - P[\phi \land \psi]\\ & = P[\psi] \end{align}$

Now do the replacement on the other side to get $P[\phi] = P[\psi]$.