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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 2807 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
3 | 2 | baib 885 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1312 ∈ wcel 1461 {crab 2392 |
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 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-rab 2397 df-v 2657 |
This theorem is referenced by: unimax 3734 undifexmid 4075 frind 4232 ordtriexmidlem2 4394 ordtriexmid 4395 ordtri2orexmid 4396 onsucelsucexmid 4403 0elsucexmid 4438 ordpwsucexmid 4443 ordtri2or2exmid 4444 acexmidlema 5717 acexmidlemb 5718 isnumi 6985 genpelvl 7262 genpelvu 7263 cauappcvgprlemladdru 7406 cauappcvgprlem1 7409 caucvgprlem1 7429 sup3exmid 8619 supinfneg 9286 infsupneg 9287 supminfex 9288 ublbneg 9301 negm 9303 hashinfuni 10410 infssuzex 11484 gcddvds 11494 dvdslegcd 11495 bezoutlemsup 11537 lcmval 11584 dvdslcm 11590 isprm2lem 11637 |
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