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Mirrors > Home > ILE Home > Th. List > uneq1 | GIF version |
Description: Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
uneq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2163 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | orbi1d 746 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) |
3 | elun 3164 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶)) | |
4 | elun 3164 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 222 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ 𝑥 ∈ (𝐵 ∪ 𝐶))) |
6 | 5 | eqrdv 2098 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 670 = wceq 1299 ∈ wcel 1448 ∪ cun 3019 |
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 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-un 3025 |
This theorem is referenced by: uneq2 3171 uneq12 3172 uneq1i 3173 uneq1d 3176 prprc1 3578 uniprg 3698 unexb 4301 relresfld 5004 relcoi1 5006 rdgeq2 6199 xpider 6430 findcard2 6712 findcard2s 6713 unfiexmid 6735 bdunexb 12699 bj-unexg 12700 |
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