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Mirrors > Home > ILE Home > Th. List > imordc | GIF version |
Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 716, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.) |
Ref | Expression |
---|---|
imordc | ⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotbdc 867 | . . 3 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑)) | |
2 | 1 | imbi1d 230 | . 2 ⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓))) |
3 | dcn 837 | . . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑) | |
4 | dfordc 887 | . . 3 ⊢ (DECID ¬ 𝜑 → ((¬ 𝜑 ∨ 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓))) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 ∨ 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓))) |
6 | 2, 5 | bitr4d 190 | 1 ⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 703 DECID wdc 829 |
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 609 ax-in2 610 ax-io 704 |
This theorem depends on definitions: df-bi 116 df-dc 830 |
This theorem is referenced by: pm4.62dc 893 pm2.26dc 902 nf4dc 1663 algcvgblem 11990 divgcdodd 12084 |
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