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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 3224 | . 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 1331 ∪ cun 3069 |
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-v 2688 df-un 3075 |
This theorem is referenced by: ifeq2 3478 tpeq3 3611 iununir 3896 unisucg 4336 relcoi1 5070 resasplitss 5302 fvun1 5487 fmptapd 5611 fvunsng 5614 fnsnsplitss 5619 tfr1onlemaccex 6245 tfrcllemaccex 6258 rdgeq1 6268 rdgivallem 6278 rdgisuc1 6281 rdgon 6283 rdg0 6284 oav2 6359 oasuc 6360 omv2 6361 omsuc 6368 fnsnsplitdc 6401 unsnfidcex 6808 undifdc 6812 fiintim 6817 ssfirab 6822 fnfi 6825 fidcenumlemr 6843 sbthlemi5 6849 sbthlemi6 6850 pm54.43 7046 fzsuc 9849 fseq1p1m1 9874 fseq1m1p1 9875 fzosplitsnm1 9986 fzosplitsn 10010 fzosplitprm1 10011 resunimafz0 10574 zfz1isolemsplit 10581 fsumm1 11185 ennnfonelemp1 11919 ennnfonelemhdmp1 11922 ennnfonelemkh 11925 ennnfonelemhf1o 11926 ennnfonelemnn0 11935 strsetsid 11992 setscom 11999 |
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