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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 2958 necon4bd 2961 noetasuplem4 27239 eulercrct 29495 notornotel1 36963 mpobi123f 37030 mptbi12f 37034 oexpreposd 41212 axfrege31 42584 clsk1independent 42797 con3ALT2 43291 zfregs2VD 43602 con3ALTVD 43677 notnotrALT2 43688 suplesup 44049 |
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