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 ↔ ( 𝜑 → 𝜓 ) ) )
2		orc	⊢ ( 𝜑 → ( 𝜑 ∨ 𝜒 ) )
3	2	biantrud	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜑 ∨ 𝜒 ) ) ) )
4		dfifp3	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜑 ∨ 𝜒 ) ) )
5	3 4	bitr4di	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , 𝜒 ) ) )
6	1 5	bitr2d	⊢ ( 𝜑 → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ 𝜓 ) )