Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > peirce | Structured version Visualization version GIF version |
Description: Peirce's axiom. This odd-looking theorem is the "difference" between an intuitionistic system of propositional calculus and a classical system and is not accepted by intuitionists. When Peirce's axiom is added to an intuitionistic system, the system becomes equivalent to our classical system ax-1 6 through ax-3 8. A notable fact about this theorem is that it requires ax-3 8 for its proof even though the result has no negation connectives in it. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 9-Oct-2012.) |
Ref | Expression |
---|---|
peirce | ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplim 161 | . 2 ⊢ (¬ (𝜑 → 𝜓) → 𝜑) | |
2 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
3 | 1, 2 | ja 171 | 1 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem is referenced by: looinv 192 tbw-ax3 1617 tb-ax3 31356 bj-peircecurry 31521 bj-peircei 31529 |
Copyright terms: Public domain | W3C validator |