Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-hadnot | Structured version Visualization version GIF version |
Description: The adder sum distributes over negation. Copy of hadnot 1603. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) |
Ref | Expression |
---|---|
wl-hadnot | ⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notbi 321 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓)) | |
2 | 1 | bibi1i 341 | . 2 ⊢ (((𝜑 ↔ 𝜓) ↔ ¬ 𝜒) ↔ ((¬ 𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜒)) |
3 | xor3 386 | . . 3 ⊢ (¬ ((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ ¬ 𝜒)) | |
4 | wl-dfhad3 34765 | . . 3 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒)) | |
5 | 3, 4 | xchnxbir 335 | . 2 ⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ ¬ 𝜒)) |
6 | wl-dfhad3 34765 | . 2 ⊢ (hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜒)) | |
7 | 2, 5, 6 | 3bitr4i 305 | 1 ⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 haddwhad 1593 |
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 209 df-an 399 df-or 844 df-ifp 1058 df-xor 1502 df-tru 1540 df-had 1594 |
This theorem is referenced by: (None) |
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