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	|- ( -. ph -> ( if- ( ph , ps , ch ) <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		ifpn	|- ( if- ( ph , ps , ch ) <-> if- ( -. ph , ch , ps ) )
2		ifptru	|- ( -. ph -> ( if- ( -. ph , ch , ps ) <-> ch ) )
3	1 2	bitrid	|- ( -. ph -> ( if- ( ph , ps , ch ) <-> ch ) )