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| Mirrors > Home > MPE Home > Th. List > sseq1 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for subclasses. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
| Ref | Expression |
|---|---|
| sseq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqss 3999 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
| 2 | sstr2 3990 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ 𝐶 → 𝐵 ⊆ 𝐶)) | |
| 3 | sstr2 3990 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶)) | |
| 4 | 2, 3 | anbiim 641 | . . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
| 5 | 4 | ancoms 458 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
| 6 | 1, 5 | sylbi 217 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
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