Theorem hadcoma

Description: Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 17-Dec-2023)

	Ref	Expression
Assertion	hadcoma	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\psi ,\phi ,\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bicom	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\psi \leftrightarrow \phi \right)$$
2	1	bibi1i	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \chi \right)\leftrightarrow \left(\left(\psi \leftrightarrow \phi \right)\leftrightarrow \chi \right)$$
3		hadbi	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \chi \right)$$
4		hadbi	$$⊢hadd\left(\psi ,\phi ,\chi \right)\leftrightarrow \left(\left(\psi \leftrightarrow \phi \right)\leftrightarrow \chi \right)$$
5	2 3 4	3bitr4i	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\psi ,\phi ,\chi \right)$$