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Mirrors > Home > ILE Home > Th. List > uneq12i | GIF version |
Description: Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
Ref | Expression |
---|---|
uneq1i.1 | ⊢ 𝐴 = 𝐵 |
uneq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
uneq12i | ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | uneq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
3 | uneq12 3189 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | |
4 | 1, 2, 3 | mp2an 420 | 1 ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷) |
Colors of variables: wff set class |
Syntax hints: = wceq 1312 ∪ cun 3033 |
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 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-v 2657 df-un 3039 |
This theorem is referenced by: indir 3289 difundir 3293 symdif1 3305 unrab 3311 rabun2 3319 dfif6 3440 dfif3 3451 unopab 3965 xpundi 4553 xpundir 4554 xpun 4558 dmun 4704 resundi 4788 resundir 4789 cnvun 4900 rnun 4903 imaundi 4907 imaundir 4908 dmtpop 4970 coundi 4996 coundir 4997 unidmrn 5027 dfdm2 5029 mptun 5210 fpr 5554 fvsnun2 5570 sbthlemi5 6799 djuunr 6901 djuun 6902 casedm 6921 djudm 6940 djuassen 7018 fzo0to42pr 9884 |
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