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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 2899 | . 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 105 = wceq 1364 ∈ wcel 2160 {cab 2175 Vcvv 2752 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 |
This theorem is referenced by: elpw 3599 elint 3868 opabid 4278 elrn2 4890 elimasn 5016 oprabid 5932 tfrlem3a 6339 tfrcllemsucaccv 6383 tfrcllembxssdm 6385 tfrcllemres 6391 addnqprlemrl 7591 addnqprlemru 7592 addnqprlemfl 7593 addnqprlemfu 7594 mulnqprlemrl 7607 mulnqprlemru 7608 mulnqprlemfl 7609 mulnqprlemfu 7610 ltnqpr 7627 ltnqpri 7628 archpr 7677 cauappcvgprlemladdfu 7688 cauappcvgprlemladdfl 7689 caucvgprlemladdfu 7711 caucvgprprlemopu 7733 suplocexprlemloc 7755 4sqlem12 12445 txuni2 14241 |
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