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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, pm2.53 722, 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 | pm2.53 722 | . 2 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) | |
2 | pm2.54dc 891 | . 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))) | |
3 | 1, 2 | impbid2 143 | 1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∨ wo 708 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-dc 835 |
This theorem is referenced by: imordc 897 pm4.64dc 900 pm5.17dc 904 pm5.6dc 926 pm3.12dc 958 pm5.15dc 1389 19.32dc 1679 r19.30dc 2624 r19.32vdc 2626 prime 9350 isprm4 12113 prm2orodd 12120 euclemma 12140 phiprmpw 12216 |
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