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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 2893 | . 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 2739 |
This theorem is referenced by: unimax 3843 undifexmid 4193 frind 4352 ordtriexmidlem2 4519 ordtriexmid 4520 ontriexmidim 4521 ordtri2orexmid 4522 onsucelsucexmid 4529 0elsucexmid 4564 ordpwsucexmid 4569 ordtri2or2exmid 4570 ontri2orexmidim 4571 canth 5828 acexmidlema 5865 acexmidlemb 5866 isnumi 7180 genpelvl 7510 genpelvu 7511 cauappcvgprlemladdru 7654 cauappcvgprlem1 7657 caucvgprlem1 7677 sup3exmid 8913 supinfneg 9594 infsupneg 9595 supminfex 9596 ublbneg 9612 negm 9614 hashinfuni 10756 infssuzex 11949 gcddvds 11963 dvdslegcd 11964 bezoutlemsup 12009 uzwodc 12037 lcmval 12062 dvdslcm 12068 isprm2lem 12115 |
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