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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 2835 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∈ wcel 1481 {cab 2126 Vcvv 2689 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 |
This theorem is referenced by: elpw 3521 elint 3785 opabid 4187 elrn2 4789 elimasn 4914 oprabid 5811 tfrlem3a 6215 tfrcllemsucaccv 6259 tfrcllembxssdm 6261 tfrcllemres 6267 addnqprlemrl 7389 addnqprlemru 7390 addnqprlemfl 7391 addnqprlemfu 7392 mulnqprlemrl 7405 mulnqprlemru 7406 mulnqprlemfl 7407 mulnqprlemfu 7408 ltnqpr 7425 ltnqpri 7426 archpr 7475 cauappcvgprlemladdfu 7486 cauappcvgprlemladdfl 7487 caucvgprlemladdfu 7509 caucvgprprlemopu 7531 suplocexprlemloc 7553 txuni2 12464 |
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