![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > notnotr | Structured version Visualization version GIF version |
Description: Double negation elimination. Converse of notnot 142 and one implication of notnotb 315. Theorem *2.14 of [WhiteheadRussell] p. 102. This was the fifth axiom of Frege, specifically Proposition 31 of [Frege1879] p. 44. In classical logic (our logic) this is always true. In intuitionistic logic this is not always true, and formulas for which it is true are called "stable". (Contributed by NM, 29-Dec-1992.) (Proof shortened by David Harvey, 5-Sep-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) |
Ref | Expression |
---|---|
notnotr | ⊢ (¬ ¬ 𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.18 128 | . 2 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑) | |
2 | 1 | jarli 126 | 1 ⊢ (¬ ¬ 𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → 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: notnotrd 133 con2d 134 con3d 152 notnotb 315 ecase3adOLD 1036 necon1ad 2961 necon4bd 2964 noetasuplem4 27100 eulercrct 29228 notornotel1 36583 mpobi123f 36650 mptbi12f 36654 oexpreposd 40836 axfrege31 42179 clsk1independent 42392 con3ALT2 42886 zfregs2VD 43197 con3ALTVD 43272 notnotrALT2 43283 suplesup 43647 |
Copyright terms: Public domain | W3C validator |