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Mirrors > Home > ILE Home > Th. List > pm5.6dc | GIF version |
Description: Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 917). (Contributed by Jim Kingdon, 2-Apr-2018.) |
Ref | Expression |
---|---|
pm5.6dc | ⊢ (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impexp 261 | . 2 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓 → 𝜒))) | |
2 | dfordc 882 | . . 3 ⊢ (DECID 𝜓 → ((𝜓 ∨ 𝜒) ↔ (¬ 𝜓 → 𝜒))) | |
3 | 2 | imbi2d 229 | . 2 ⊢ (DECID 𝜓 → ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → (¬ 𝜓 → 𝜒)))) |
4 | 1, 3 | bitr4id 198 | 1 ⊢ (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 DECID wdc 824 |
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 604 ax-in2 605 ax-io 699 |
This theorem depends on definitions: df-bi 116 df-dc 825 |
This theorem is referenced by: (None) |
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