Theorem wl-df2-3mintru2

Description: The adder carry in disjunctive normal form. An alternative highly symmetric definition emphasizing the independence 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 ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		andi	⊢ ( ( 𝜑 ∧ ( 𝜓 ∨ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ) )
2	1	orbi1i	⊢ ( ( ( 𝜑 ∧ ( 𝜓 ∨ 𝜒 ) ) ∨ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ) ∨ ( 𝜓 ∧ 𝜒 ) ) )
3		wl-df-3mintru2	⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ if- ( 𝜑 , ( 𝜓 ∨ 𝜒 ) , ( 𝜓 ∧ 𝜒 ) ) )
4		animorl	⊢ ( ( 𝜓 ∧ 𝜒 ) → ( 𝜓 ∨ 𝜒 ) )
5		wl-ifpimpr	⊢ ( ( ( 𝜓 ∧ 𝜒 ) → ( 𝜓 ∨ 𝜒 ) ) → ( if- ( 𝜑 , ( 𝜓 ∨ 𝜒 ) , ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ ( 𝜓 ∨ 𝜒 ) ) ∨ ( 𝜓 ∧ 𝜒 ) ) ) )
6	4 5	ax-mp	⊢ ( if- ( 𝜑 , ( 𝜓 ∨ 𝜒 ) , ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ ( 𝜓 ∨ 𝜒 ) ) ∨ ( 𝜓 ∧ 𝜒 ) ) )
7	3 6	bitri	⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ ( 𝜓 ∨ 𝜒 ) ) ∨ ( 𝜓 ∧ 𝜒 ) ) )
8		df-3or	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ) ∨ ( 𝜓 ∧ 𝜒 ) ) )
9	2 7 8	3bitr4i	⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) )