Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > pm2.6dc | GIF version |
Description: Case elimination for a decidable proposition. Based on theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 25-Mar-2018.) |
Ref | Expression |
---|---|
pm2.6dc | ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.1dc 745 | . . 3 ⊢ (DECID 𝜑 → (¬ 𝜑 ∨ 𝜑)) | |
2 | pm3.44 635 | . . 3 ⊢ (((¬ 𝜑 → 𝜓) ∧ (𝜑 → 𝜓)) → ((¬ 𝜑 ∨ 𝜑) → 𝜓)) | |
3 | 1, 2 | syl5com 26 | . 2 ⊢ (DECID 𝜑 → (((¬ 𝜑 → 𝜓) ∧ (𝜑 → 𝜓)) → 𝜓)) |
4 | 3 | expd 245 | 1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∨ wo 629 DECID wdc 742 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 |
This theorem is referenced by: jadc 760 jaddc 761 pm2.61dc 762 |
Copyright terms: Public domain | W3C validator |