Theorem ifpdfan2

Description: Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020)

	Ref	Expression
Assertion	ifpdfan2	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow if-\left(\phi ,\psi ,\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢\phi \to \phi$$
2	1	notnoti	$$⊢\neg \neg \left(\phi \to \phi \right)$$
3	2	biorfi	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\vee \neg \left(\phi \to \phi \right)\right)$$
4		dfifp6	$$⊢if-\left(\phi ,\psi ,\phi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\vee \neg \left(\phi \to \phi \right)\right)$$
5	3 4	bitr4i	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow if-\left(\phi ,\psi ,\phi \right)$$