Theorem wl-df3-3mintru2

Description: The adder carry in conjunctive normal form. An alternative highly symmetric definition emphasizing the independence of order of the inputs ph , ps and ch . Copy of cadan . (Contributed by Mario Carneiro, 4-Sep-2016) df-cad redefined. (Revised by Wolf Lammen, 18-Jun-2024)

	Ref	Expression
Assertion	wl-df3-3mintru2	|- ( cadd ( ph , ps , ch ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		ordi	|- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) )
2	1	anbi1i	|- ( ( ( 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-ifp4impr	|- ( ( ( 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-3an	|- ( ( ( 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 ) ) )