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 ( 𝜑 → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 biimt ( 𝜑 → ( 𝜓 ↔ ( 𝜑𝜓 ) ) )
2 orc ( 𝜑 → ( 𝜑𝜒 ) )
3 2 biantrud ( 𝜑 → ( ( 𝜑𝜓 ) ↔ ( ( 𝜑𝜓 ) ∧ ( 𝜑𝜒 ) ) ) )
4 dfifp3 ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑𝜓 ) ∧ ( 𝜑𝜒 ) ) )
5 3 4 bitr4di ( 𝜑 → ( ( 𝜑𝜓 ) ↔ if- ( 𝜑 , 𝜓 , 𝜒 ) ) )
6 1 5 bitr2d ( 𝜑 → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ 𝜓 ) )