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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 3298 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷) |
Colors of variables: wff set class |
Syntax hints: = 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: indir 3398 difundir 3402 symdif1 3414 unrab 3420 rabun2 3428 dfif6 3550 dfif3 3561 unopab 4096 xpundi 4696 xpundir 4697 xpun 4701 dmun 4848 resundi 4934 resundir 4935 cnvun 5048 rnun 5051 imaundi 5055 imaundir 5056 dmtpop 5118 coundi 5144 coundir 5145 unidmrn 5175 dfdm2 5177 mptun 5361 fpr 5713 fvsnun2 5729 sbthlemi5 6977 djuunr 7082 djuun 7083 casedm 7102 djudm 7121 djuassen 7233 fz0to3un2pr 10140 fz0to4untppr 10141 fzo0to42pr 10237 |
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