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Mirrors > Home > ILE Home > Th. List > sseq1d | GIF version |
Description: An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
sseq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
sseq1d | ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | sseq1 3070 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1299 ⊆ wss 3021 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-11 1452 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-in 3027 df-ss 3034 |
This theorem is referenced by: sseq12d 3078 eqsstrd 3083 snssg 3603 ssiun2s 3804 treq 3972 onsucsssucexmid 4380 funimass1 5136 feq1 5191 sbcfg 5207 fvmptssdm 5437 fvimacnvi 5466 nnsucsssuc 6318 ereq1 6366 elpm2r 6490 nnnninf 6935 ctssexmid 6936 iscnp 12149 iscnp4 12168 cnntr 12175 cnconst2 12183 cnptopresti 12188 cnptoprest 12189 txbas 12208 txcnp 12221 txdis 12227 txdis1cn 12228 blssps 12355 blss 12356 ssblex 12359 blin2 12360 metss2 12426 metrest 12434 metcnp3 12435 limccl 12510 ellimc3ap 12511 |
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