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Mirrors > Home > ILE Home > Th. List > elrab3 | GIF version |
Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.) |
Ref | Expression |
---|---|
elrab.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elrab3 | ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrab.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | elrab 2813 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
3 | 2 | baib 889 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 {crab 2397 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rab 2402 df-v 2662 |
This theorem is referenced by: unimax 3740 undifexmid 4087 frind 4244 ordtriexmidlem2 4406 ordtriexmid 4407 ordtri2orexmid 4408 onsucelsucexmid 4415 0elsucexmid 4450 ordpwsucexmid 4455 ordtri2or2exmid 4456 acexmidlema 5733 acexmidlemb 5734 isnumi 7006 genpelvl 7288 genpelvu 7289 cauappcvgprlemladdru 7432 cauappcvgprlem1 7435 caucvgprlem1 7455 sup3exmid 8683 supinfneg 9358 infsupneg 9359 supminfex 9360 ublbneg 9373 negm 9375 hashinfuni 10491 infssuzex 11569 gcddvds 11579 dvdslegcd 11580 bezoutlemsup 11624 lcmval 11671 dvdslcm 11677 isprm2lem 11724 |
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