Equivalent conditions for perfect secrecy of a symmetric crypto system

I've been reading about perfect secrecy in crypto systems and I've ran across two definitions which turn out to be equivalent.

The first is Shannon secrecy:

A crypto system $(\cal K, \cal M$, $\text{Gen, Enc, Dec})$ is said to have Shannon secrecy if for all distributions $\cal D$ over $\cal M$ and for all $m\in\cal M, c\in \cal C$ $Pr_K[M=m| C=c]=Pr_K[M=m]$

where $K,M,C$ are random variables whose distribution is given by the distribution on $\cal M, \cal K$.

The second is Perfect secrecy:

A crypto system $(\cal K, \cal M$, $\text{Gen, Enc, Dec})$ is said to have perfect secrecy if for all $m_1,m_2\in\cal M, c\in \cal C$ $Pr_K[Enc(K,m_1)=c]=Pr_K[Enc(K,m_2)=c]$

My question is if that is also equivalent to the following:

for all $c_1,c_2 \in \cal C, m\in \cal M$:

$Pr_K[Dec(K,c_1)=m]=Pr_K[Dec(K,c_2)=m]$

Thanks in advance.