| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > sseqtrd | Structured version Visualization version GIF version | ||
| Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) |
| Ref | Expression |
|---|---|
| sseqtrd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| sseqtrd.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| sseqtrd | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sseqtrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 2 | sseqtrd.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
| 3 | 2 | sseq2d 4016 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)) |
| 4 | 1, 3 | mpbid 232 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
| Copyright terms: Public domain | W3C validator |