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Mirrors > Home > ILE Home > Th. List > eqsstrid | GIF version |
Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
eqsstrid.1 | ⊢ 𝐴 = 𝐵 |
eqsstrid.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
eqsstrid | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstrid.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
2 | eqsstrid.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 2 | sseq1i 3154 | . 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶) |
4 | 1, 3 | sylibr 133 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ⊆ wss 3102 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-in 3108 df-ss 3115 |
This theorem is referenced by: eqsstrrid 3175 inss 3337 difsnss 3702 tpssi 3722 peano5 4556 xpsspw 4697 iotanul 5149 iotass 5151 fun 5341 fun11iun 5434 fvss 5481 fmpt 5616 fliftrel 5739 opabbrex 5862 1stcof 6108 2ndcof 6109 tfrlemibacc 6270 tfrlemibfn 6272 tfr1onlemssrecs 6283 tfr1onlembacc 6286 tfr1onlembfn 6288 tfrcllemssrecs 6296 tfrcllembacc 6299 tfrcllembfn 6301 caucvgprlemladdrl 7593 peano5nnnn 7807 peano5nni 8831 un0addcl 9118 un0mulcl 9119 strleund 12265 cnptopco 12609 cnconst2 12620 xmetresbl 12827 blsscls2 12880 bj-omtrans 13518 |
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