Description: Commutative law for the adders sum. (Contributed by Mario Carneiro, 4-Sep-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | hadcomb | |- ( hadd ( ph , ps , ch ) <-> hadd ( ph , ch , ps ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biid | |- ( ph <-> ph ) |
|
2 | xorcom | |- ( ( ps \/_ ch ) <-> ( ch \/_ ps ) ) |
|
3 | 1 2 | xorbi12i | |- ( ( ph \/_ ( ps \/_ ch ) ) <-> ( ph \/_ ( ch \/_ ps ) ) ) |
4 | hadass | |- ( hadd ( ph , ps , ch ) <-> ( ph \/_ ( ps \/_ ch ) ) ) |
|
5 | hadass | |- ( hadd ( ph , ch , ps ) <-> ( ph \/_ ( ch \/_ ps ) ) ) |
|
6 | 3 4 5 | 3bitr4i | |- ( hadd ( ph , ps , ch ) <-> hadd ( ph , ch , ps ) ) |