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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 2966 | . 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 98 = wceq 1243 ⊆ wss 2917 |
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 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 |
This theorem is referenced by: sseq12d 2974 eqsstrd 2979 snssg 3500 ssiun2s 3701 treq 3860 onsucsssucexmid 4252 funimass1 4976 feq1 5030 sbcfg 5045 fvmptssdm 5255 fvimacnvi 5281 nnsucsssuc 6071 ereq1 6113 |
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