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Mirrors > Home > ILE Home > Th. List > elab2 | GIF version |
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |
Ref | Expression |
---|---|
elab2.1 | ⊢ 𝐴 ∈ V |
elab2.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
elab2.3 | ⊢ 𝐵 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
elab2 | ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elab2.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elab2.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | elab2.3 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
4 | 2, 3 | elab2g 2750 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
5 | 1, 4 | ax-mp 7 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1285 ∈ wcel 1434 {cab 2069 Vcvv 2612 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 |
This theorem is referenced by: elpw 3412 elint 3668 opabid 4048 elrn2 4635 elimasn 4754 oprabid 5616 tfrlem3a 6007 tfrcllemsucaccv 6051 tfrcllembxssdm 6053 tfrcllemres 6059 addnqprlemrl 7019 addnqprlemru 7020 addnqprlemfl 7021 addnqprlemfu 7022 mulnqprlemrl 7035 mulnqprlemru 7036 mulnqprlemfl 7037 mulnqprlemfu 7038 ltnqpr 7055 ltnqpri 7056 archpr 7105 cauappcvgprlemladdfu 7116 cauappcvgprlemladdfl 7117 caucvgprlemladdfu 7139 caucvgprprlemopu 7161 |
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