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Incoming links: Subproof
- The implication introduction rule assumes locally, derives under that assumption, then concludes while discharging the temporary assumption.
- The disjunction elimination rule uses and two local cases: one derives a common conclusion from , and the other derives that same from . It concludes and discharges both case assumptions.
- The negation introduction rule assumes locally and derives logical falsity. Discharging that assumption gives .
The boxes delimit a subproof and identify the lifetime of its temporary assumptions. A result depending on a boxed assumption is available inside that box and its nested boxes; it cannot simply be exported outside. The enclosing rule specifies what may be exported and which assumptions are discharged. Other premises on which the derivation depends remain undischarged. This assumption discharge discipline distinguishes a conditional argument from an unconditional proof.
Derive each negated component in its own subproof, then combine them. Assumption lines below begin local scopes; the listed discharge ranges close those scopes.
| Line | Formula | Justification |
|---|---|---|
| 1 | Premise | |
| 2 | Assume locally | |
| 3 | Disjunction introduction rule, left, 2 | |
| 4 | Negation elimination rule, 1 and 3 | |
| 5 | Negation introduction rule, discharging 2–4 | |
| 6 | Assume locally | |
| 7 | Disjunction introduction rule, right, 6 | |
| 8 | Negation elimination rule, 1 and 7 | |
| 9 | Negation introduction rule, discharging 6–8 | |
| 10 | Conjunction introduction rule, 5 and 9 |
Only the original premise remains open. No classical principle is needed for this direction of De Morgan's laws.
In a finite domain , a true existential premise says that at least one satisfies . The subproof treats its witness as an unknown object named , derives without relying on which object was chosen, and can therefore be applied to the actual witness. Thus follows whenever the existential premise is true. This argument does not require knowing the domain's size or selecting a particular named element in advance.
The existential quantifier elimination rule in sequent form records the same assumption discharge: a derivation of yields . The natural-deduction box represents the temporary witness assumption that the sequent rule replaces by the existential premise.
The eigenvariable condition requires that have no free occurrence in the other undischarged assumptions, the existential premise or the exported conclusion . Bound occurrences can be renamed harmlessly. This freshness condition keeps from denoting a previously special object and prevents witness escape. Without it, one could export from , falsely asserting that an arbitrarily named object has the witness property. A two-element domain with only one element satisfying supplies an immediate counterexample.