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Mirrors > Home > ILE Home > Th. List > inteqd | GIF version |
Description: Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.) |
Ref | Expression |
---|---|
inteqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
inteqd | ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inteqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | inteq 3782 | . 2 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∩ cint 3779 |
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-ral 2422 df-int 3780 |
This theorem is referenced by: intprg 3812 op1stbg 4408 onsucmin 4431 elreldm 4773 elxp5 5035 fniinfv 5487 1stval2 6061 2ndval2 6062 fundmen 6708 xpsnen 6723 fiintim 6825 elfi2 6868 fi0 6871 cardcl 7054 isnumi 7055 cardval3ex 7058 carden2bex 7062 clsfval 12309 clsval 12319 |
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