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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 3297 | . 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 1363 ∪ cun 3141 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2170 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-v 2753 df-un 3147 |
This theorem is referenced by: ifeq2 3552 tpeq3 3694 iununir 3984 unisucg 4428 relcoi1 5174 resasplitss 5409 fvun1 5597 fmptapd 5722 fvunsng 5725 fnsnsplitss 5730 tfr1onlemaccex 6366 tfrcllemaccex 6379 rdgeq1 6389 rdgivallem 6399 rdgisuc1 6402 rdgon 6404 rdg0 6405 oav2 6481 oasuc 6482 omv2 6483 omsuc 6490 fnsnsplitdc 6523 unsnfidcex 6936 undifdc 6940 fiintim 6945 ssfirab 6950 fnfi 6953 fidcenumlemr 6971 sbthlemi5 6977 sbthlemi6 6978 pm54.43 7206 fzsuc 10086 fseq1p1m1 10111 fseq1m1p1 10112 fzosplitsnm1 10226 fzosplitsn 10250 fzosplitprm1 10251 resunimafz0 10828 zfz1isolemsplit 10835 fsumm1 11441 fprodm1 11623 ennnfonelemp1 12424 ennnfonelemhdmp1 12427 ennnfonelemkh 12430 ennnfonelemhf1o 12431 ennnfonelemnn0 12440 strsetsid 12512 setscom 12519 lspun0 13701 bj-charfundcALT 14944 |
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