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Mirrors > Home > MPE Home > Th. List > ineq12 | Structured version Visualization version GIF version |
Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.) |
Ref | Expression |
---|---|
ineq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 4203 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) | |
2 | ineq2 4204 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∩ 𝐶) = (𝐵 ∩ 𝐷)) | |
3 | 1, 2 | sylan9eq 2785 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∩ cin 3943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-in 3951 |
This theorem is referenced by: ineq12i 4208 ineq12d 4211 ineqan12d 4212 fnun 6669 frrlem4 8295 undifixp 8953 endisj 9083 sbthlem8 9115 fiin 9447 pm54.43 10026 kmlem9 10183 indistopon 22948 epttop 22956 restbas 23106 ordtbas2 23139 txbas 23515 ptbasin 23525 trfbas2 23791 snfil 23812 fbasrn 23832 trfil2 23835 fmfnfmlem3 23904 ustuqtop2 24191 minveclem3b 25400 isperp 28588 brredunds 38228 diophin 42334 kelac2lem 42630 iscnrm3r 48153 |
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