Theorem had1

Description: If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 11-Jul-2020)

	Ref	Expression
Assertion	had1	$$⊢\phi \to \left(hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		hadrot	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\psi ,\chi ,\phi \right)$$
2		hadbi	$$⊢hadd\left(\psi ,\chi ,\phi \right)\leftrightarrow \left(\left(\psi \leftrightarrow \chi \right)\leftrightarrow \phi \right)$$
3	1 2	bitri	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\psi \leftrightarrow \chi \right)\leftrightarrow \phi \right)$$
4		biass	$$⊢\left(\left(hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)\leftrightarrow \phi \right)\leftrightarrow \left(hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\psi \leftrightarrow \chi \right)\leftrightarrow \phi \right)\right)$$
5	3 4	mpbir	$$⊢\left(hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)\leftrightarrow \phi$$
6	5	biimpri	$$⊢\phi \to \left(hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)$$