Description: If the first input is true, then triple xor is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | wl-3xortru | |- ( ph -> ( hadd ( ph , ps , ch ) <-> -. ( ps \/_ ch ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wl-df-3xor | |- ( hadd ( ph , ps , ch ) <-> if- ( ph , -. ( ps \/_ ch ) , ( ps \/_ ch ) ) ) |
|
| 2 | ifptru | |- ( ph -> ( if- ( ph , -. ( ps \/_ ch ) , ( ps \/_ ch ) ) <-> -. ( ps \/_ ch ) ) ) |
|
| 3 | 1 2 | syl5bb | |- ( ph -> ( hadd ( ph , ps , ch ) <-> -. ( ps \/_ ch ) ) ) |