  The Vop\v{e}nka principle, which asserts that every proper class of
first-order structures in a common language admits an elementary embedding
between two of its members, is not equivalent over GBC to the first-order
Vop\v{e}nka scheme, which makes the Vop\v{e}nka assertion only for the
first-order definable classes of structures. Nevertheless, the two Vop\v{e}nka
axioms are equiconsistent and they have exactly the same first-order
consequences in the language of set theory. Specifically, GBC plus the
Vop\v{e}nka principle is conservative over ZFC plus the Vop\v{e}nka scheme for
first-order assertions in the language of set theory.
