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 ) ) |
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 ) ) |