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Mirrors > Home > ILE Home > Th. List > pm2.18dc | GIF version |
Description: Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called Clavius law). Intuitionistically it requires a decidability assumption, but compare with pm2.01 616 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.) |
Ref | Expression |
---|---|
pm2.18dc | ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.21 617 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 → ¬ (¬ 𝜑 → 𝜑))) | |
2 | 1 | a2i 11 | . . 3 ⊢ ((¬ 𝜑 → 𝜑) → (¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) |
3 | condc 853 | . . 3 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑)) → ((¬ 𝜑 → 𝜑) → 𝜑))) | |
4 | 2, 3 | syl5 32 | . 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑))) |
5 | 4 | pm2.43d 50 | 1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 DECID wdc 834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 |
This theorem is referenced by: pm4.81dc 908 |
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