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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 3774 | . 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 1331 ∩ cint 3771 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-int 3772 |
This theorem is referenced by: intprg 3804 op1stbg 4400 onsucmin 4423 elreldm 4765 elxp5 5027 fniinfv 5479 1stval2 6053 2ndval2 6054 fundmen 6700 xpsnen 6715 fiintim 6817 elfi2 6860 fi0 6863 cardcl 7037 isnumi 7038 cardval3ex 7041 carden2bex 7045 clsfval 12270 clsval 12280 |
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