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| Mirrors > Home > MPE Home > Th. List > 3sstr4d | Structured version Visualization version GIF version | ||
| Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
| Ref | Expression |
|---|---|
| 3sstr4d.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| 3sstr4d.2 | ⊢ (𝜑 → 𝐶 = 𝐴) |
| 3sstr4d.3 | ⊢ (𝜑 → 𝐷 = 𝐵) |
| Ref | Expression |
|---|---|
| 3sstr4d | ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3sstr4d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐴) | |
| 2 | 3sstr4d.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 3 | 1, 2 | eqsstrd 4018 | . 2 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
| 4 | 3sstr4d.3 | . 2 ⊢ (𝜑 → 𝐷 = 𝐵) | |
| 5 | 3, 4 | sseqtrrd 4021 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
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