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Mirrors > Home > ILE Home > Th. List > uneq12d | GIF version |
Description: Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
uneq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
uneq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
uneq12d | ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | uneq12d.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | uneq12 3195 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | |
4 | 1, 2, 3 | syl2anc 408 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∪ cun 3039 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 |
This theorem is referenced by: disjpr2 3557 diftpsn3 3631 iunxprg 3863 undifexmid 4087 exmidundif 4099 exmidundifim 4100 suceq 4294 rnpropg 4988 fntpg 5149 foun 5354 fnimapr 5449 fprg 5571 fsnunfv 5589 fsnunres 5590 tfrlemi1 6197 tfr1onlemaccex 6213 tfrcllemaccex 6226 ereq1 6404 undifdc 6780 unfiin 6782 djueq12 6892 fztp 9826 fzsuc2 9827 fseq1p1m1 9842 ennnfonelemg 11843 ennnfonelemp1 11846 ennnfonelem1 11847 ennnfonelemnn0 11862 setsvalg 11916 setsfun0 11922 setsresg 11924 setsslid 11936 exmid1stab 13122 |
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