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| Mirrors > Home > MPE Home > Th. List > sseq12d | Structured version Visualization version GIF version | ||
| Description: An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.) |
| Ref | Expression |
|---|---|
| sseq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| sseq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| sseq12d | ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sseq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | 1 | sseq1d 4015 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
| 3 | sseq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 4 | 3 | sseq2d 4016 | . 2 ⊢ (𝜑 → (𝐵 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) |
| 5 | 2, 4 | bitrd 279 | 1 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) |
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