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Mirrors > Home > ILE Home > Th. List > notnotrdc | GIF version |
Description: Double negation elimination for a decidable proposition. The converse, notnot 591, holds for all propositions, not just decidable ones. This is Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability condition added. (Contributed by Jim Kingdon, 11-Mar-2018.) |
Ref | Expression |
---|---|
notnotrdc | ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 776 | . . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | orcom 679 | . . 3 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜑)) | |
3 | 1, 2 | bitri 182 | . 2 ⊢ (DECID 𝜑 ↔ (¬ 𝜑 ∨ 𝜑)) |
4 | pm2.53 673 | . 2 ⊢ ((¬ 𝜑 ∨ 𝜑) → (¬ ¬ 𝜑 → 𝜑)) | |
5 | 3, 4 | sylbi 119 | 1 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 661 DECID wdc 775 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 df-dc 776 |
This theorem is referenced by: dcimpstab 785 notnotbdc 799 condandc 808 pm2.13dc 812 pm2.54dc 823 |
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