How To Use Logical proof In A Sentence

Researchers have produced human embryos containing DNA from three people, a biotechnological proof-of-principle with profound medical and ethical implications. 3-Parent Embryo | Impact Lab
The camp provided the Germans tautological proof of the necessity of imprisoning Untermenschen: the disgusting state of the Haftlinge of Auschwitz demonstrated their inferiority and justified their murder.
Various cases of supernumerary testicles have been reported during operations or in physical examination (without histological proof, however).
The ontological and cosmological proofs are for those who wish to cleanse themselves of the "muddiness and accidentality" of the world. William James, part 7: Agnosticism and the will to believe
Anselm presented a ontological proof, which was re-emphasized by Descartes who, starting from first principles, proved the existence of God in a philosophical way.

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Hence the logical proof schema which enables you to deduce any proposition whatever from a contradiction cannot be applied.
The second is that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of those of logic.