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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 3230 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∪ cun 3074 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 |
This theorem is referenced by: disjpr2 3595 diftpsn3 3669 iunxprg 3901 undifexmid 4125 exmidundif 4137 exmidundifim 4138 suceq 4332 rnpropg 5026 fntpg 5187 foun 5394 fnimapr 5489 fprg 5611 fsnunfv 5629 fsnunres 5630 tfrlemi1 6237 tfr1onlemaccex 6253 tfrcllemaccex 6266 ereq1 6444 undifdc 6820 unfiin 6822 djueq12 6932 fztp 9889 fzsuc2 9890 fseq1p1m1 9905 ennnfonelemg 11952 ennnfonelemp1 11955 ennnfonelem1 11956 ennnfonelemnn0 11971 setsvalg 12028 setsfun0 12034 setsresg 12036 setsslid 12048 exmid1stab 13368 |
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