Theorem cadcoma

Description: Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016)

	Ref	Expression
Assertion	cadcoma	⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ cadd ( 𝜓 , 𝜑 , 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ancom	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) )
2		xorcom	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( 𝜓 ⊻ 𝜑 ) )
3	2	anbi2i	⊢ ( ( 𝜒 ∧ ( 𝜑 ⊻ 𝜓 ) ) ↔ ( 𝜒 ∧ ( 𝜓 ⊻ 𝜑 ) ) )
4	1 3	orbi12i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ ( 𝜑 ⊻ 𝜓 ) ) ) ↔ ( ( 𝜓 ∧ 𝜑 ) ∨ ( 𝜒 ∧ ( 𝜓 ⊻ 𝜑 ) ) ) )
5		df-cad	⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ ( 𝜑 ⊻ 𝜓 ) ) ) )
6		df-cad	⊢ ( cadd ( 𝜓 , 𝜑 , 𝜒 ) ↔ ( ( 𝜓 ∧ 𝜑 ) ∨ ( 𝜒 ∧ ( 𝜓 ⊻ 𝜑 ) ) ) )
7	4 5 6	3bitr4i	⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ cadd ( 𝜓 , 𝜑 , 𝜒 ) )