Biconditional: backward direction

Identifier: F:logic-iff-mpr
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

If p if-and-only-if q, and q holds, then p holds.

Formal statement

theorem Iff.mpr : ((x0 : Prop) -> ((x1 : Prop) -> ((x2 : Iff x0 x1) -> ((x3 : x1) -> x0))))

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

0 direct dependencies 19 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-logic-iff-mpr

Kind: kernel-term
Status: checked

Supports: If p if-and-only-if q, and q holds, then p holds.

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- Iff.mpr 2>/dev/null | grep -Ec '^Iff\.mpr[[:space:]]')" -ge 1

Evidence notes

`build_logic_prelude` (built first by `build_nat_prelude`, which `nat_theorem_inventory` constructs) admits `Iff.mpr` only through the trusted kernel gate, so this line in the tool's output IS the type-check. Anchored with `grep -Ec '^Iff.mpr[[:space:]]'` on the tool's own substring-filtered output, matching exactly one row; a name that does not exist makes both the tool's argument filter and the anchored grep fail (verified both ways 2026-08-25).

footprint-logic-iff-mpr

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- the logic environment admits no trusted declaration

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free logic

Evidence notes

Reports `logic: axiom=0 opaque=0 quotient=0 total_trusted=0`, over the full trusted surface (`Axiom` + `Opaque` + `Quotient`), not `Declaration::Axiom` alone. The enumeration is per-environment, so it bounds this theorem's footprint: a proof admitted into an environment cannot depend on a trusted declaration the environment does not contain. `logic` is the isolated, intuitionistic core (`build_logic_prelude` alone) that `nat` nests -- measured 2026-08-25 with `nat_axiom_inventory --require-axiom-free logic`.

Provenance

{
  "date": "2026-08-25",
  "established_by": "nat-axiom-coverage lane, axeyum-lean-kernel logic prelude",
  "source": "read off the kernel by `nat_theorem_inventory`, not transcribed; statement authored here",
  "prior_art": [
    {
      "citation": "Standard intuitionistic/classical propositional logic (natural deduction); no single primary source.",
      "establishes": "Standard rule for the biconditional (modus ponens on the backward implication).",
      "attribution": "standard textbook material; this lane did not consult a primary source"
    }
  ]
}