This question about pairwise vs. mutual relations is related some extant questions: here and here.
Kobayashi, Mark & Turin's Probability, Random Processes and Statistical Analysis, 2012, states without proof:
three events, A, B, C are mutually independent when:
P[A,B]=P[A]P[B], P[B,C]=P[B]P[C], P[A,C]=P[A]P[C], P[A,B,C]=P[A]P[B]P[C]
No three of these relations necessarily imply the fourth. [my italics]
However, Wikipedia and others generally agree that mutual independence implies pairwise independence, but also without a demonstration.
What is the simplest proof that mutual independence implies pairwise independence?
Note: GC Rota wrote that probability can be understood by focusing on random variables or focusing on distributions. However, the two views should be equivalent, correct?
Mutual independence means the four identities you copied, pairwise independence means the first three of these identities. Ergo.