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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-consensus | Structured version Visualization version GIF version |
Description: Version of consensus 1050 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1050, using only binary connectives, and hinting at the fact that it is a Boolean algebra identity, like the absorption identities.) (Contributed by BJ, 30-Sep-2019.) |
Ref | Expression |
---|---|
bj-consensus | ⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anifp 1069 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) | |
2 | 1 | bj-jaoi2 34753 | . 2 ⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) → if-(𝜑, 𝜓, 𝜒)) |
3 | orc 864 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) → (if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒))) | |
4 | 2, 3 | impbii 208 | 1 ⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∨ wo 844 if-wif 1060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 |
This theorem is referenced by: (None) |
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