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| Mirrors > Home > MPE Home > Th. List > sylan9eqr | Structured version Visualization version GIF version | ||
| Description: An equality transitivity deduction. (Contributed by NM, 8-May-1994.) |
| Ref | Expression |
|---|---|
| sylan9eqr.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| sylan9eqr.2 | ⊢ (𝜓 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| sylan9eqr | ⊢ ((𝜓 ∧ 𝜑) → 𝐴 = 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sylan9eqr.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | sylan9eqr.2 | . . 3 ⊢ (𝜓 → 𝐵 = 𝐶) | |
| 3 | 1, 2 | sylan9eq 2797 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
| 4 | 3 | ancoms 458 | 1 ⊢ ((𝜓 ∧ 𝜑) → 𝐴 = 𝐶) |
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