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Mirrors > Home > ILE Home > Th. List > ineq12i | GIF version |
Description: Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
Ref | Expression |
---|---|
ineq1i.1 | ⊢ 𝐴 = 𝐵 |
ineq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
ineq12i | ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | ineq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
3 | ineq12 3277 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) | |
4 | 1, 2, 3 | mp2an 423 | 1 ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷) |
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: undir 3331 difindir 3336 inrab 3353 inrab2 3354 inxp 4681 resindi 4842 resindir 4843 cnvin 4954 rnin 4956 inimass 4963 funtp 5184 imainlem 5212 imain 5213 offres 6041 djuinr 6956 djuin 6957 casefun 6978 exmidfodomrlemim 7074 enq0enq 7263 explecnv 11306 |
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