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Mirrors > Home > ILE Home > Th. List > syl5eqss | GIF version |
Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
syl5eqss.1 | ⊢ 𝐴 = 𝐵 |
syl5eqss.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
syl5eqss | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl5eqss.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
2 | syl5eqss.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 2 | sseq1i 2969 | . 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶) |
4 | 1, 3 | sylibr 137 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = 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: syl5eqssr 2990 inss 3166 difsnss 3510 tpssi 3530 peano5 4321 xpsspw 4450 iotanul 4882 iotass 4884 fun 5063 fun11iun 5147 fvss 5189 fmpt 5319 fliftrel 5432 opabbrex 5549 1stcof 5790 2ndcof 5791 tfrlemibacc 5940 tfrlemibfn 5942 caucvgprlemladdrl 6776 peano5nnnn 6966 peano5nni 7917 un0addcl 8215 un0mulcl 8216 peano5setOLD 10065 bj-omtrans 10081 |
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