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| Mirrors > Home > MPE Home > Th. List > eqsstrid | Structured version Visualization version GIF version | ||
| Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
| Ref | Expression |
|---|---|
| eqsstrid.1 | ⊢ 𝐴 = 𝐵 |
| eqsstrid.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
| Ref | Expression |
|---|---|
| eqsstrid | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqsstrid.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
| 2 | eqsstrid.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
| 3 | 2 | sseq1i 4012 | . 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶) |
| 4 | 1, 3 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
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