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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 2963 | . 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 1242 ⊆ wss 2911 |
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 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 |
This theorem is referenced by: syl5eqssr 2984 inss 3160 difsnss 3501 tpssi 3521 peano5 4264 xpsspw 4393 iotanul 4825 iotass 4827 fun 5006 fun11iun 5090 fvss 5132 fmpt 5262 fliftrel 5375 opabbrex 5491 1stcof 5732 2ndcof 5733 tfrlemibacc 5881 tfrlemibfn 5883 caucvgprlemladdrl 6649 peano5nni 7698 un0addcl 7991 un0mulcl 7992 peano5setOLD 9400 bj-omtrans 9416 |
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