Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dfordc | GIF version |
Description: Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 711, 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 | pm2.53 711 | . 2 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) | |
2 | pm2.54dc 876 | . 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))) | |
3 | 1, 2 | impbid2 142 | 1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 697 DECID wdc 819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 |
This theorem depends on definitions: df-bi 116 df-dc 820 |
This theorem is referenced by: imordc 882 pm4.64dc 885 pm5.17dc 889 pm5.6dc 911 pm3.12dc 942 pm5.15dc 1367 19.32dc 1657 r19.32vdc 2580 prime 9150 isprm4 11800 prm2orodd 11807 euclemma 11824 phiprmpw 11898 |
Copyright terms: Public domain | W3C validator |