Description: If one case of an if- condition is a consequence of the other, the expression in dfifp4 can be shortened. (Contributed by Wolf Lammen, 18-Jun-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | wl-ifp4impr | ⊢ ( ( 𝜒 → 𝜓 ) → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜒 ) ∧ 𝜓 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-ifpimpr | ⊢ ( ( 𝜒 → 𝜓 ) → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ 𝜒 ) ) ) | |
2 | pm4.71 | ⊢ ( ( 𝜒 → 𝜓 ) ↔ ( 𝜒 ↔ ( 𝜒 ∧ 𝜓 ) ) ) | |
3 | 2 | biimpi | ⊢ ( ( 𝜒 → 𝜓 ) → ( 𝜒 ↔ ( 𝜒 ∧ 𝜓 ) ) ) |
4 | 3 | orbi2d | ⊢ ( ( 𝜒 → 𝜓 ) → ( ( ( 𝜑 ∧ 𝜓 ) ∨ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ 𝜓 ) ) ) ) |
5 | andir | ⊢ ( ( ( 𝜑 ∨ 𝜒 ) ∧ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ 𝜓 ) ) ) | |
6 | 4 5 | bitr4di | ⊢ ( ( 𝜒 → 𝜓 ) → ( ( ( 𝜑 ∧ 𝜓 ) ∨ 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜒 ) ∧ 𝜓 ) ) ) |
7 | 1 6 | bitrd | ⊢ ( ( 𝜒 → 𝜓 ) → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜒 ) ∧ 𝜓 ) ) ) |