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A Boolean function satisfying the displayed condition. Composition of one-preserving Boolean functions is one-preserving, since each inner output is one on an all-one input. Consequently structural induction shows that a variable-built propositional formula using only one-preserving logical connectives, such as $\wedge ,\vee ,\to$, is true under the all-one Boolean valuation.

Use structural induction on the propositional formula. The induction claim is that its truth value under the all-one Boolean valuation is one. A variable has this truth value by definition. For the induction step, suppose the immediate subformulas $\alpha ,\beta$ each have truth value one. ThenThese are all permitted construction cases. Thus every such propositional formula evaluates to one. Equivalently, the three logical connectives define one-preserving Boolean functions, and composition preserves this property. The variable-only base grammar matters: admitting a constant false formula would invalidate the claim.