Metamath Proof Explorer


Theorem had1

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 ) ) )

Proof

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 ) ) )