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Mirrors > Home > ILE Home > Th. List > elab2g | GIF version |
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |
Ref | Expression |
---|---|
elab2g.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
elab2g.2 | ⊢ 𝐵 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
elab2g | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elab2g.2 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
2 | 1 | eleq2i 2206 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
3 | elab2g.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 3 | elabg 2830 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
5 | 2, 4 | syl5bb 191 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {cab 2125 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 |
This theorem is referenced by: elab2 2832 elab4g 2833 eldif 3080 elun 3217 elin 3259 elsng 3542 elprg 3547 eluni 3739 eliun 3817 eliin 3818 elopab 4180 elong 4295 opeliunxp 4594 elrn2g 4729 eldmg 4734 elrnmpt 4788 elrnmpt1 4790 elimag 4885 elrnmpog 5883 eloprabi 6094 tfrlem3ag 6206 tfr1onlem3ag 6234 tfrcllemsucaccv 6251 elqsg 6479 elixp2 6596 isomni 7008 ismkv 7027 1idprl 7398 1idpru 7399 recexprlemell 7430 recexprlemelu 7431 mertenslemub 11303 mertenslemi1 11304 mertenslem2 11305 istopg 12166 isbasisg 12211 |
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