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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 2826 | . 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 1331 ∈ wcel 1480 {cab 2123 Vcvv 2681 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 |
This theorem is referenced by: elpw 3511 elint 3772 opabid 4174 elrn2 4776 elimasn 4901 oprabid 5796 tfrlem3a 6200 tfrcllemsucaccv 6244 tfrcllembxssdm 6246 tfrcllemres 6252 addnqprlemrl 7358 addnqprlemru 7359 addnqprlemfl 7360 addnqprlemfu 7361 mulnqprlemrl 7374 mulnqprlemru 7375 mulnqprlemfl 7376 mulnqprlemfu 7377 ltnqpr 7394 ltnqpri 7395 archpr 7444 cauappcvgprlemladdfu 7455 cauappcvgprlemladdfl 7456 caucvgprlemladdfu 7478 caucvgprprlemopu 7500 suplocexprlemloc 7522 txuni2 12414 |
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