Description: Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 11-May-1993) (Proof shortened by Wolf Lammen, 19-May-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | imbid.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
| Assertion | bibi2d | ⊢ ( 𝜑 → ( ( 𝜃 ↔ 𝜓 ) ↔ ( 𝜃 ↔ 𝜒 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imbid.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
| 2 | 1 | pm5.74i | ⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) |
| 3 | 2 | bibi2i | ⊢ ( ( ( 𝜑 → 𝜃 ) ↔ ( 𝜑 → 𝜓 ) ) ↔ ( ( 𝜑 → 𝜃 ) ↔ ( 𝜑 → 𝜒 ) ) ) |
| 4 | pm5.74 | ⊢ ( ( 𝜑 → ( 𝜃 ↔ 𝜓 ) ) ↔ ( ( 𝜑 → 𝜃 ) ↔ ( 𝜑 → 𝜓 ) ) ) | |
| 5 | pm5.74 | ⊢ ( ( 𝜑 → ( 𝜃 ↔ 𝜒 ) ) ↔ ( ( 𝜑 → 𝜃 ) ↔ ( 𝜑 → 𝜒 ) ) ) | |
| 6 | 3 4 5 | 3bitr4i | ⊢ ( ( 𝜑 → ( 𝜃 ↔ 𝜓 ) ) ↔ ( 𝜑 → ( 𝜃 ↔ 𝜒 ) ) ) |
| 7 | 6 | pm5.74ri | ⊢ ( 𝜑 → ( ( 𝜃 ↔ 𝜓 ) ↔ ( 𝜃 ↔ 𝜒 ) ) ) |