Theorem cadcoma

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

	Ref	Expression
Assertion	cadcoma	$$⊢cadd\left(\phi ,\psi ,\chi \right)\leftrightarrow cadd\left(\psi ,\phi ,\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ancom	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \left(\psi \wedge \phi \right)$$
2		xorcom	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \left(\psi ⊻\phi \right)$$
3	2	anbi2i	$$⊢\left(\chi \wedge \left(\phi ⊻\psi \right)\right)\leftrightarrow \left(\chi \wedge \left(\psi ⊻\phi \right)\right)$$
4	1 3	orbi12i	$$⊢\left(\left(\phi \wedge \psi \right)\vee \left(\chi \wedge \left(\phi ⊻\psi \right)\right)\right)\leftrightarrow \left(\left(\psi \wedge \phi \right)\vee \left(\chi \wedge \left(\psi ⊻\phi \right)\right)\right)$$
5		df-cad	$$⊢cadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\vee \left(\chi \wedge \left(\phi ⊻\psi \right)\right)\right)$$
6		df-cad	$$⊢cadd\left(\psi ,\phi ,\chi \right)\leftrightarrow \left(\left(\psi \wedge \phi \right)\vee \left(\chi \wedge \left(\psi ⊻\phi \right)\right)\right)$$
7	4 5 6	3bitr4i	$$⊢cadd\left(\phi ,\psi ,\chi \right)\leftrightarrow cadd\left(\psi ,\phi ,\chi \right)$$