Description: If the first input is false, then the adder sum is equivalent to the exclusive disjunction of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 12-Jul-2020)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | had0 | ⊢ ( ¬ 𝜑 → ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜓 ⊻ 𝜒 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | had1 | ⊢ ( ¬ 𝜑 → ( hadd ( ¬ 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) ↔ ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) ) | |
| 2 | hadnot | ⊢ ( ¬ hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ hadd ( ¬ 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) ) | |
| 3 | xnor | ⊢ ( ( 𝜓 ↔ 𝜒 ) ↔ ¬ ( 𝜓 ⊻ 𝜒 ) ) | |
| 4 | notbi | ⊢ ( ( 𝜓 ↔ 𝜒 ) ↔ ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) | |
| 5 | 3 4 | bitr3i | ⊢ ( ¬ ( 𝜓 ⊻ 𝜒 ) ↔ ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) |
| 6 | 1 2 5 | 3bitr4g | ⊢ ( ¬ 𝜑 → ( ¬ hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ¬ ( 𝜓 ⊻ 𝜒 ) ) ) |
| 7 | 6 | con4bid | ⊢ ( ¬ 𝜑 → ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜓 ⊻ 𝜒 ) ) ) |