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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 3530 tpeq3 3671 iununir 3956 unisucg 4399 relcoi1 5142 resasplitss 5377 fvun1 5562 fmptapd 5687 fvunsng 5690 fnsnsplitss 5695 tfr1onlemaccex 6327 tfrcllemaccex 6340 rdgeq1 6350 rdgivallem 6360 rdgisuc1 6363 rdgon 6365 rdg0 6366 oav2 6442 oasuc 6443 omv2 6444 omsuc 6451 fnsnsplitdc 6484 unsnfidcex 6897 undifdc 6901 fiintim 6906 ssfirab 6911 fnfi 6914 fidcenumlemr 6932 sbthlemi5 6938 sbthlemi6 6939 pm54.43 7167 fzsuc 10025 fseq1p1m1 10050 fseq1m1p1 10051 fzosplitsnm1 10165 fzosplitsn 10189 fzosplitprm1 10190 resunimafz0 10766 zfz1isolemsplit 10773 fsumm1 11379 fprodm1 11561 ennnfonelemp1 12361 ennnfonelemhdmp1 12364 ennnfonelemkh 12367 ennnfonelemhf1o 12368 ennnfonelemnn0 12377 strsetsid 12449 setscom 12456 bj-charfundcALT 13844 |
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