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Mirrors > Home > ILE Home > Th. List > uneq2d | GIF version |
Description: Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.) |
Ref | Expression |
---|---|
uneq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
uneq2d | ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | uneq2 3275 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∪ cun 3119 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 |
This theorem is referenced by: ifeq2 3529 tpeq3 3669 iununir 3954 unisucg 4397 relcoi1 5140 resasplitss 5375 fvun1 5560 fmptapd 5684 fvunsng 5687 fnsnsplitss 5692 tfr1onlemaccex 6324 tfrcllemaccex 6337 rdgeq1 6347 rdgivallem 6357 rdgisuc1 6360 rdgon 6362 rdg0 6363 oav2 6439 oasuc 6440 omv2 6441 omsuc 6448 fnsnsplitdc 6481 unsnfidcex 6893 undifdc 6897 fiintim 6902 ssfirab 6907 fnfi 6910 fidcenumlemr 6928 sbthlemi5 6934 sbthlemi6 6935 pm54.43 7154 fzsuc 10012 fseq1p1m1 10037 fseq1m1p1 10038 fzosplitsnm1 10152 fzosplitsn 10176 fzosplitprm1 10177 resunimafz0 10753 zfz1isolemsplit 10760 fsumm1 11366 fprodm1 11548 ennnfonelemp1 12348 ennnfonelemhdmp1 12351 ennnfonelemkh 12354 ennnfonelemhf1o 12355 ennnfonelemnn0 12364 strsetsid 12436 setscom 12443 bj-charfundcALT 13766 |
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