Description: If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 11-Jul-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | had1 | |- ( ph -> ( hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hadrot | |- ( hadd ( ph , ps , ch ) <-> hadd ( ps , ch , ph ) ) |
|
2 | hadbi | |- ( hadd ( ps , ch , ph ) <-> ( ( ps <-> ch ) <-> ph ) ) |
|
3 | 1 2 | bitri | |- ( hadd ( ph , ps , ch ) <-> ( ( ps <-> ch ) <-> ph ) ) |
4 | biass | |- ( ( ( hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) <-> ph ) <-> ( hadd ( ph , ps , ch ) <-> ( ( ps <-> ch ) <-> ph ) ) ) |
|
5 | 3 4 | mpbir | |- ( ( hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) <-> ph ) |
6 | 5 | biimpri | |- ( ph -> ( hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) ) |