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The class forcing theorem, which asserts that every class forcing notion 
${\mathbb {P}}$
admits a forcing relation 
$\Vdash _{\mathbb {P}}$
, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory 
$\text {GBC}$
to the principle of elementary transfinite recursion 
$\text {ETR}_{\text {Ord}}$
for class recursions of length 
$\text {Ord}$
. It is also equivalent to the existence of truth predicates for the infinitary languages 
$\mathcal {L}_{\text {Ord},\omega }(\in ,A)$
, allowing any class parameter A; to the existence of truth predicates for the language 
$\mathcal {L}_{\text {Ord},\text {Ord}}(\in ,A)$
; to the existence of 
$\text {Ord}$
-iterated truth predicates for first-order set theory 
$\mathcal {L}_{\omega ,\omega }(\in ,A)$
; to the assertion that every separative class partial order 
${\mathbb {P}}$
has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most 
$\text {Ord}+1$
. Unlike set forcing, if every class forcing notion 
${\mathbb {P}}$
has a forcing relation merely for atomic formulas, then every such 
${\mathbb {P}}$
has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between 
$\text {GBC}$
and Kelley–Morse set theory 
$\text {KM}$
.
