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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 3116 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ⊆ wss 3066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-in 3072 df-ss 3079 |
This theorem is referenced by: sseqtri 3126 sseqtrdi 3140 abss 3161 ssrab 3170 ssintrab 3789 iunpwss 3899 iotass 5100 dffun2 5128 ssimaex 5475 isstructim 11962 isstructr 11963 bj-ssom 13123 ss1oel2o 13178 |
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