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Mirrors > Home > ILE Home > Th. List > sseq2d | 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 |
---|---|
sseq2d | ⊢ (𝜑 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | sseq2 3121 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ⊆ wss 3071 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: sseq12d 3128 sseqtrd 3135 exmidsssn 4125 exmidsssnc 4126 onsucsssucexmid 4442 sbcrel 4625 funimass2 5201 fnco 5231 fnssresb 5235 fnimaeq0 5244 foimacnv 5385 fvelimab 5477 ssimaexg 5483 fvmptss2 5496 rdgss 6280 summodclem2 11158 summodc 11159 zsumdc 11160 fsum3cvg3 11172 prodmodclem2 11353 prodmodc 11354 ennnfoneleminc 11931 isbasisg 12221 tgval 12228 tgss3 12257 restbasg 12347 tgrest 12348 restopn2 12362 cnpnei 12398 cnptopresti 12417 txbas 12437 elmopn 12625 neibl 12670 dvfgg 12836 |
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