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Mirrors > Home > ILE Home > Th. List > inteq | GIF version |
Description: Equality law for intersection. (Contributed by NM, 13-Sep-1999.) |
Ref | Expression |
---|---|
inteq | ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq 2555 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)) | |
2 | 1 | abbidv 2200 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} = {𝑥 ∣ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦}) |
3 | dfint2 3659 | . 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} | |
4 | dfint2 3659 | . 2 ⊢ ∩ 𝐵 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦} | |
5 | 2, 3, 4 | 3eqtr4g 2140 | 1 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 {cab 2069 ∀wral 2353 ∩ cint 3657 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-int 3658 |
This theorem is referenced by: inteqi 3661 inteqd 3662 uniintsnr 3693 rint0 3696 intexr 3946 onintexmid 4344 elreldm 4609 elxp5 4860 1stval2 5835 fundmen 6376 xpsnen 6388 bj-intexr 10991 |
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