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Mirrors > Home > ILE Home > Th. List > dfor2dc | GIF version |
Description: Logical 'or' expressed in terms of implication only, for a decidable proposition. Based on theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 27-Mar-2018.) |
Ref | Expression |
---|---|
dfor2dc | ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 → 𝜓) → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.62 699 | . 2 ⊢ ((𝜑 ∨ 𝜓) → ((𝜑 → 𝜓) → 𝜓)) | |
2 | pm2.68dc 826 | . 2 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓))) | |
3 | 1, 2 | impbid2 141 | 1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 → 𝜓) → 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∨ wo 661 DECID wdc 775 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 df-dc 776 |
This theorem is referenced by: imimorbdc 828 |
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