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Mirrors > Home > ILE Home > Th. List > ineq2 | GIF version |
Description: Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.) |
Ref | Expression |
---|---|
ineq2 | ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 3240 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) | |
2 | incom 3238 | . 2 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶) | |
3 | incom 3238 | . 2 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶) | |
4 | 1, 2, 3 | 3eqtr4g 2175 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∩ cin 3040 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 |
This theorem is referenced by: ineq12 3242 ineq2i 3244 ineq2d 3247 uneqin 3297 intprg 3774 fiintim 6785 uzin2 10727 inopn 12097 basis1 12141 basis2 12142 baspartn 12144 metreslem 12476 qtopbasss 12617 |
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