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Mirrors > Home > ILE Home > Th. List > sseq2i | GIF version |
Description: An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
sseq1i.1 | ⊢ A = B |
Ref | Expression |
---|---|
sseq2i | ⊢ (𝐶 ⊆ A ↔ 𝐶 ⊆ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1i.1 | . 2 ⊢ A = B | |
2 | sseq2 2961 | . 2 ⊢ (A = B → (𝐶 ⊆ A ↔ 𝐶 ⊆ B)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐶 ⊆ A ↔ 𝐶 ⊆ B) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1242 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 |
This theorem is referenced by: sseqtri 2971 syl6sseq 2985 abss 3003 ssrab 3012 ssintrab 3629 iunpwss 3734 iotass 4827 dffun2 4855 ssimaex 5177 bj-ssom 9395 |
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