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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-df2-3mintru2 | Structured version Visualization version GIF version |
Description: The adder carry in disjunctive normal form. An alternative highly symmetric definition emphasizing the independance of order of the inputs 𝜑, 𝜓 and 𝜒. Copy of cador 1611. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 12-Jun-2024.) |
Ref | Expression |
---|---|
wl-df2-3mintru2 | ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | andi 1006 | . . 3 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒))) | |
2 | 1 | orbi1i 912 | . 2 ⊢ (((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜓 ∧ 𝜒)) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒))) |
3 | wl-df-3mintru2 35166 | . . 3 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒))) | |
4 | animorl 976 | . . . 4 ⊢ ((𝜓 ∧ 𝜒) → (𝜓 ∨ 𝜒)) | |
5 | wl-ifpimpr 35148 | . . . 4 ⊢ (((𝜓 ∧ 𝜒) → (𝜓 ∨ 𝜒)) → (if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜓 ∧ 𝜒)))) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ (if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜓 ∧ 𝜒))) |
7 | 3, 6 | bitri 278 | . 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜓 ∧ 𝜒))) |
8 | df-3or 1086 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒))) | |
9 | 2, 7, 8 | 3bitr4i 307 | 1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 845 if-wif 1059 ∨ w3o 1084 caddwcad 1609 |
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 210 df-an 401 df-or 846 df-ifp 1060 df-3or 1086 df-3an 1087 df-xor 1504 df-cad 1610 |
This theorem is referenced by: wl-df4-3mintru2 35169 |
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