A new scheme for proving pseudoidentities from a given set {Sigma} of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when {Sigma} defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples when the scheme is complete are given when {Sigma} defines a pseudovariety V which is {sigma}-reducible for the equation x=y, provided {Sigma} is enough to prove a basis of identities for the variety of {sigma}-algebras generated by V. This gives ample evidence in support of the conjecture that the proof scheme is complete in general.
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