Theorem ifptru

Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue . This is essentially dedlema . (Contributed by BJ, 20-Sep-2019) (Proof shortened by Wolf Lammen, 10-Jul-2020)

	Ref	Expression
Assertion	ifptru	|- ( ph -> ( if- ( ph , ps , ch ) <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		biimt	|- ( ph -> ( ps <-> ( ph -> ps ) ) )
2		orc	|- ( ph -> ( ph \/ ch ) )
3	2	biantrud	|- ( ph -> ( ( ph -> ps ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) ) )
4		dfifp3	|- ( if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) )
5	3 4	bitr4di	|- ( ph -> ( ( ph -> ps ) <-> if- ( ph , ps , ch ) ) )
6	1 5	bitr2d	|- ( ph -> ( if- ( ph , ps , ch ) <-> ps ) )