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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 3092 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | |
4 | 1, 2, 3 | syl2anc 391 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∪ cun 2915 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 |
This theorem is referenced by: disjpr2 3434 diftpsn3 3505 suceq 4139 rnpropg 4800 fntpg 4955 foun 5145 fnimapr 5233 fprg 5346 fsnunfv 5363 fsnunres 5364 tfrlemi1 5946 ereq1 6113 fztp 8940 fzsuc2 8941 fseq1p1m1 8956 |
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