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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 3276 | . 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 1332 ∩ cin 3075 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 |
This theorem is referenced by: in4 3297 inindir 3299 indif2 3325 difun1 3341 dfrab3ss 3359 dfif3 3492 intunsn 3817 rint0 3818 riin0 3892 res0 4831 resres 4839 resundi 4840 resindi 4842 inres 4844 resiun2 4847 resopab 4871 dfse2 4920 dminxp 4991 imainrect 4992 resdmres 5038 funimacnv 5207 unfiin 6822 sbthlemi5 6857 dmaddpi 7157 dmmulpi 7158 fsumiun 11278 |
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