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Mirrors > Home > ILE Home > Th. List > ineq1d | GIF version |
Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.) |
Ref | Expression |
---|---|
ineq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
ineq1d | ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | ineq1 3275 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = 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: diftpsn3 3669 disji2 3930 ordpwsucexmid 4493 riinint 4808 fnresdisj 5241 fnimadisj 5251 ecinxp 6512 fiintim 6825 fival 6866 fzval2 9824 fvinim0ffz 10049 fsum1p 11219 restopnb 12389 metrest 12714 qtopbasss 12729 |
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