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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 3862 | . 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 1364 ∩ cint 3859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-int 3860 |
This theorem is referenced by: intprg 3892 op1stbg 4497 onsucmin 4524 elreldm 4871 elxp5 5135 fniinfv 5595 1stval2 6181 2ndval2 6182 fundmen 6833 xpsnen 6848 fiintim 6958 elfi2 7002 fi0 7005 cardcl 7211 isnumi 7212 cardval3ex 7215 carden2bex 7219 lspfval 13721 lspval 13723 lsppropd 13765 clsfval 14078 clsval 14088 |
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