| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > sylan9eq | Structured version Visualization version GIF version | ||
| Description: An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
| Ref | Expression |
|---|---|
| sylan9eq.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| sylan9eq.2 | ⊢ (𝜓 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| sylan9eq | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sylan9eq.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | sylan9eq.2 | . 2 ⊢ (𝜓 → 𝐵 = 𝐶) | |
| 3 | eqtr 2760 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶) | |
| 4 | 1, 2, 3 | syl2an 596 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
| Copyright terms: Public domain | W3C validator |