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Mirrors > Home > MPE Home > Th. List > inteq | Structured version Visualization version 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 3115 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)) | |
2 | 1 | abbidv 2728 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} = {𝑥 ∣ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦}) |
3 | dfint2 4412 | . 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} | |
4 | dfint2 4412 | . 2 ⊢ ∩ 𝐵 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦} | |
5 | 2, 3, 4 | 3eqtr4g 2669 | 1 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 {cab 2596 ∀wral 2896 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-int 4411 |
This theorem is referenced by: inteqi 4414 inteqd 4415 unissint 4436 uniintsn 4449 rint0 4452 intex 4747 intnex 4748 elreldm 5271 elxp5 7004 1stval2 7076 oev2 7490 fundmen 7916 xpsnen 7929 fiint 8122 elfir 8204 inelfi 8207 fiin 8211 cardmin2 8707 isfin2-2 9024 incexclem 14407 mreintcl 16078 ismred2 16086 fiinopn 20531 cmpfii 21022 ptbasfi 21194 fbssint 21452 shintcl 27573 chintcl 27575 inelpisys 29544 rankeq1o 31448 neificl 32719 heibor1lem 32778 elrfi 36275 elrfirn 36276 |
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