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 ) ) |
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 ) ) |