Metamath Proof Explorer


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 ) )