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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 2894 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
3 | 2 | baib 919 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 {crab 2459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-v 2740 |
This theorem is referenced by: unimax 3844 undifexmid 4194 frind 4353 ordtriexmidlem2 4520 ordtriexmid 4521 ontriexmidim 4522 ordtri2orexmid 4523 onsucelsucexmid 4530 0elsucexmid 4565 ordpwsucexmid 4570 ordtri2or2exmid 4571 ontri2orexmidim 4572 canth 5829 acexmidlema 5866 acexmidlemb 5867 isnumi 7181 genpelvl 7511 genpelvu 7512 cauappcvgprlemladdru 7655 cauappcvgprlem1 7658 caucvgprlem1 7678 sup3exmid 8914 supinfneg 9595 infsupneg 9596 supminfex 9597 ublbneg 9613 negm 9615 hashinfuni 10757 infssuzex 11950 gcddvds 11964 dvdslegcd 11965 bezoutlemsup 12010 uzwodc 12038 lcmval 12063 dvdslcm 12069 isprm2lem 12116 |
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