A formal proof is a finite sequence of formulae in which every line is an axiom, an assumption, or follows from earlier lines by a specified inference rule.
A conclusion is formally derivable from premises when a permitted formal proof exists with those premises as its undischarged assumptions. This is a syntactic relation, distinct from semantic entailment.
Theoremhood is formal derivability with no undischarged assumptions. A logical formula is a theorem of a calculus exactly when it has an assumption-free formal proof.
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A formal proof is a rigorous mathematical demonstration that establishes the truth of a statement or theorem through a series of logical deductions based on agreed-upon axioms and inference rules. Formal proofs are characterized by their strict adherence to a defined formal system, which includes: 1. **Axioms**: Fundamental statements or propositions assumed to be true without proof. They serve as the starting point for any arguments or proofs.
A proof in some system for the formalization of mathematics.