Idempotent Mal'cev conditions and 2-uniform congruences (Q1001457)

In a previous paper [Algebra Univers. 57, No.~1, 63--73 (2007; Zbl 1121.08002)], the author showed that in finite algebras with a majority term any two 2-uniform congruences permute. Now he proves that the existence of a majority term can be weakened to the requirement that some nontrivial idempotent Mal'tsev condition be satisfied in the sense of \textit{G. Grätzer} [J. Comb. Theory 8, 334--342 (1970; Zbl 0194.01401)], which is equivalent to the existence of a ``Taylor'' term, as in Corollary 5.3 of \textit{W. Taylor} [Can. J. Math. 29, 498--527 (1977; Zbl 0357.08004)]. The logical structure of the present proof is somewhat irritating.