Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > pm5.12dc | GIF version |
Description: Excluded middle with antecedents for a decidable consequent. Based on theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.) |
Ref | Expression |
---|---|
pm5.12dc | ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 743 | . 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓)) | |
2 | ax-1 5 | . . 3 ⊢ (𝜓 → (𝜑 → 𝜓)) | |
3 | ax-1 5 | . . 3 ⊢ (¬ 𝜓 → (𝜑 → ¬ 𝜓)) | |
4 | 2, 3 | orim12i 676 | . 2 ⊢ ((𝜓 ∨ ¬ 𝜓) → ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓))) |
5 | 1, 4 | sylbi 114 | 1 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ 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: (None) |
Copyright terms: Public domain | W3C validator |