Theorem hadbi

Description: The adder sum is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016)

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

Proof

Step	Hyp	Ref	Expression
1		df-xor	$$⊢\left(\left(\phi ⊻\psi \right)⊻\chi \right)\leftrightarrow \neg \left(\left(\phi ⊻\psi \right)\leftrightarrow \chi \right)$$
2		df-had	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi ⊻\psi \right)⊻\chi \right)$$
3		xnor	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\phi ⊻\psi \right)$$
4	3	bibi1i	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \chi \right)\leftrightarrow \left(\neg \left(\phi ⊻\psi \right)\leftrightarrow \chi \right)$$
5		nbbn	$$⊢\left(\neg \left(\phi ⊻\psi \right)\leftrightarrow \chi \right)\leftrightarrow \neg \left(\left(\phi ⊻\psi \right)\leftrightarrow \chi \right)$$
6	4 5	bitri	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \chi \right)\leftrightarrow \neg \left(\left(\phi ⊻\psi \right)\leftrightarrow \chi \right)$$
7	1 2 6	3bitr4i	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \chi \right)$$