Metamath Proof Explorer

Theorem wl-df2-3mintru2

Description: The adder carry in disjunctive normal form. An alternative highly symmetric definition emphasizing the independance of order of the inputs ph , ps and ch . Copy of cador . (Contributed by Mario Carneiro, 4-Sep-2016) df-cad redefined. (Revised by Wolf Lammen, 12-Jun-2024)

Ref Expression
Assertion wl-df2-3mintru2 
|- ( cadd ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) \/ ( ps /\ ch ) ) )

Proof

Step Hyp Ref Expression
1 andi 
 |-  ( ( ph /\ ( ps \/ ch ) ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
2 1 orbi1i 
 |-  ( ( ( ph /\ ( ps \/ ch ) ) \/ ( ps /\ ch ) ) <-> ( ( ( ph /\ ps ) \/ ( ph /\ ch ) ) \/ ( ps /\ ch ) ) )
3 wl-df-3mintru2 
 |-  ( cadd ( ph , ps , ch ) <-> if- ( ph , ( ps \/ ch ) , ( ps /\ ch ) ) )
4 animorl 
 |-  ( ( ps /\ ch ) -> ( ps \/ ch ) )
5 wl-ifpimpr 
 |-  ( ( ( ps /\ ch ) -> ( ps \/ ch ) ) -> ( if- ( ph , ( ps \/ ch ) , ( ps /\ ch ) ) <-> ( ( ph /\ ( ps \/ ch ) ) \/ ( ps /\ ch ) ) ) )
6 4 5 ax-mp 
 |-  ( if- ( ph , ( ps \/ ch ) , ( ps /\ ch ) ) <-> ( ( ph /\ ( ps \/ ch ) ) \/ ( ps /\ ch ) ) )
7 3 6 bitri 
 |-  ( cadd ( ph , ps , ch ) <-> ( ( ph /\ ( ps \/ ch ) ) \/ ( ps /\ ch ) ) )
8 df-3or 
 |-  ( ( ( ph /\ ps ) \/ ( ph /\ ch ) \/ ( ps /\ ch ) ) <-> ( ( ( ph /\ ps ) \/ ( ph /\ ch ) ) \/ ( ps /\ ch ) ) )
9 2 7 8 3bitr4i 
 |-  ( cadd ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) \/ ( ps /\ ch ) ) )