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Mirrors > Home > MPE Home > Th. List > sseqtrid | Structured version Visualization version GIF version |
Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
sseqtrid.1 | ⊢ 𝐵 ⊆ 𝐴 |
sseqtrid.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
Ref | Expression |
---|---|
sseqtrid | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqtrid.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
2 | sseqtrid.1 | . 2 ⊢ 𝐵 ⊆ 𝐴 | |
3 | sseq2 3948 | . . 3 ⊢ (𝐴 = 𝐶 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐶)) | |
4 | 3 | biimpa 477 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐶) |
5 | 1, 2, 4 | sylancl 586 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
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