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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 836, 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 805 | . . 3 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑)) | |
2 | 1 | imbi1d 230 | . 2 ⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓))) |
3 | dcn 785 | . . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑) | |
4 | dfordc 830 | . . 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 665 DECID wdc 781 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 |
This theorem depends on definitions: df-bi 116 df-dc 782 |
This theorem is referenced by: pm4.62dc 837 pm2.26dc 852 nf4dc 1606 algcvgblem 11370 divgcdodd 11461 |
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