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 | |- ( -. ph -> ( hadd ( ph , ps , ch ) <-> ( ps \/_ ch ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | had1 | |- ( -. ph -> ( hadd ( -. ph , -. ps , -. ch ) <-> ( -. ps <-> -. ch ) ) ) |
|
| 2 | hadnot | |- ( -. hadd ( ph , ps , ch ) <-> hadd ( -. ph , -. ps , -. ch ) ) |
|
| 3 | xnor | |- ( ( ps <-> ch ) <-> -. ( ps \/_ ch ) ) |
|
| 4 | notbi | |- ( ( ps <-> ch ) <-> ( -. ps <-> -. ch ) ) |
|
| 5 | 3 4 | bitr3i | |- ( -. ( ps \/_ ch ) <-> ( -. ps <-> -. ch ) ) |
| 6 | 1 2 5 | 3bitr4g | |- ( -. ph -> ( -. hadd ( ph , ps , ch ) <-> -. ( ps \/_ ch ) ) ) |
| 7 | 6 | con4bid | |- ( -. ph -> ( hadd ( ph , ps , ch ) <-> ( ps \/_ ch ) ) ) |