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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 3190 | . 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 1314 ∪ cun 3035 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 |
This theorem is referenced by: ifeq2 3444 tpeq3 3577 iununir 3862 unisucg 4296 relcoi1 5028 resasplitss 5260 fvun1 5441 fmptapd 5565 fvunsng 5568 fnsnsplitss 5573 tfr1onlemaccex 6199 tfrcllemaccex 6212 rdgeq1 6222 rdgivallem 6232 rdgisuc1 6235 rdgon 6237 rdg0 6238 oav2 6313 oasuc 6314 omv2 6315 omsuc 6322 fnsnsplitdc 6355 unsnfidcex 6761 undifdc 6765 fiintim 6770 ssfirab 6774 fnfi 6777 fidcenumlemr 6795 sbthlemi5 6801 sbthlemi6 6802 pm54.43 6996 fzsuc 9742 fseq1p1m1 9767 fseq1m1p1 9768 fzosplitsnm1 9879 fzosplitsn 9903 fzosplitprm1 9904 resunimafz0 10467 zfz1isolemsplit 10474 fsumm1 11077 ennnfonelemp1 11764 ennnfonelemhdmp1 11767 ennnfonelemkh 11770 ennnfonelemhf1o 11771 ennnfonelemnn0 11780 strsetsid 11835 setscom 11842 |
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