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Mirrors > Home > ILE Home > Th. List > ineq1 | GIF version |
Description: Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.) |
Ref | Expression |
---|---|
ineq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2204 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | anbi1d 461 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) |
3 | elin 3264 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶)) | |
4 | elin 3264 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 222 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ 𝑥 ∈ (𝐵 ∩ 𝐶))) |
6 | 5 | eqrdv 2138 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 ∩ 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: ineq2 3276 ineq12 3277 ineq1i 3278 ineq1d 3281 dfrab3ss 3359 intprg 3812 inex1g 4072 reseq1 4821 fiintim 6825 uzin2 10791 elrestr 12167 inopn 12209 isbasisg 12250 basis1 12253 basis2 12254 tgval 12257 ntrfval 12308 tgrest 12377 restco 12382 restsn 12388 restopnb 12389 txrest 12484 metrest 12714 qtopbasss 12729 bdinex1g 13270 |
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