Description: Obsolete version of bian1d as of 29-Jun-2025. (Contributed by Thierry Arnoux, 26-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | bian1d.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) ) | |
| Assertion | bian1dOLD | ⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜃 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bian1d.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) ) | |
| 2 | 1 | biimpd | ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∧ 𝜃 ) ) ) |
| 3 | 2 | adantld | ⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) → ( 𝜒 ∧ 𝜃 ) ) ) |
| 4 | simpl | ⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜒 ) | |
| 5 | 4 | a1i | ⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜃 ) → 𝜒 ) ) |
| 6 | 1 | biimprd | ⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜃 ) → 𝜓 ) ) |
| 7 | 5 6 | jcad | ⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜃 ) → ( 𝜒 ∧ 𝜓 ) ) ) |
| 8 | 3 7 | impbid | ⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜃 ) ) ) |