Theorem ifpfal

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse . This is essentially dedlemb . (Contributed by BJ, 20-Sep-2019) (Proof shortened by Wolf Lammen, 25-Jun-2020)

	Ref	Expression
Assertion	ifpfal	$$⊢\neg \phi \to \left(if-\left(\phi ,\psi ,\chi \right)\leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ifpn	$$⊢if-\left(\phi ,\psi ,\chi \right)\leftrightarrow if-\left(\neg \phi ,\chi ,\psi \right)$$
2		ifptru	$$⊢\neg \phi \to \left(if-\left(\neg \phi ,\chi ,\psi \right)\leftrightarrow \chi \right)$$
3	1 2	syl5bb	$$⊢\neg \phi \to \left(if-\left(\phi ,\psi ,\chi \right)\leftrightarrow \chi \right)$$