| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > 3bitr4i | Structured version Visualization version GIF version | ||
| Description: A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) |
| Ref | Expression |
|---|---|
| 3bitr4i.1 | ⊢ (𝜑 ↔ 𝜓) |
| 3bitr4i.2 | ⊢ (𝜒 ↔ 𝜑) |
| 3bitr4i.3 | ⊢ (𝜃 ↔ 𝜓) |
| Ref | Expression |
|---|---|
| 3bitr4i | ⊢ (𝜒 ↔ 𝜃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3bitr4i.2 | . 2 ⊢ (𝜒 ↔ 𝜑) | |
| 2 | 3bitr4i.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
| 3 | 3bitr4i.3 | . . 3 ⊢ (𝜃 ↔ 𝜓) | |
| 4 | 2, 3 | bitr4i 278 | . 2 ⊢ (𝜑 ↔ 𝜃) |
| 5 | 1, 4 | bitri 275 | 1 ⊢ (𝜒 ↔ 𝜃) |
| Copyright terms: Public domain | W3C validator |