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Mirrors > Home > ILE Home > Th. List > orimdidc | GIF version |
Description: Disjunction distributes over implication. The forward direction, pm2.76 754, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 844. (Contributed by Jim Kingdon, 1-Apr-2018.) |
Ref | Expression |
---|---|
orimdidc | ⊢ (DECID 𝜑 → ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.76 754 | . 2 ⊢ ((𝜑 ∨ (𝜓 → 𝜒)) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))) | |
2 | pm2.85dc 844 | . 2 ⊢ (DECID 𝜑 → (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒)))) | |
3 | 1, 2 | impbid2 141 | 1 ⊢ (DECID 𝜑 → ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))) |
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-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 df-dc 776 |
This theorem is referenced by: orbididc 894 |
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