Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dfordc | GIF version |
Description: Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 641, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.) |
Ref | Expression |
---|---|
dfordc | ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.53 641 | . 2 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) | |
2 | pm2.54dc 790 | . 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))) | |
3 | 1, 2 | impbid2 131 | 1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∨ 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-in1 544 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 |
This theorem is referenced by: imordc 796 pm4.64dc 801 pm5.17dc 810 pm5.6dc 835 pm3.12dc 865 pm5.15dc 1280 19.32dc 1569 r19.32vdc 2459 prime 8337 |
Copyright terms: Public domain | W3C validator |