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Mirrors > Home > ILE Home > Th. List > ineq2i | GIF version |
Description: Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.) |
Ref | Expression |
---|---|
ineq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
ineq2i | ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | ineq2 3322 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∩ cin 3120 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 |
This theorem is referenced by: in4 3343 inindir 3345 indif2 3371 difun1 3387 dfrab3ss 3405 dfif3 3539 intunsn 3869 rint0 3870 riin0 3944 res0 4895 resres 4903 resundi 4904 resindi 4906 inres 4908 resiun2 4911 resopab 4935 dfse2 4984 dminxp 5055 imainrect 5056 resdmres 5102 funimacnv 5274 unfiin 6903 sbthlemi5 6938 dmaddpi 7287 dmmulpi 7288 fsumiun 11440 |
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