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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 2256 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
3 | elab2g.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 3 | elabg 2898 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
5 | 2, 4 | bitrid 192 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 {cab 2175 |
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: elab2 2900 elab4g 2901 eldif 3153 elun 3291 elin 3333 elsng 3622 elprg 3627 eluni 3827 eliun 3905 eliin 3906 elopab 4276 elong 4391 opeliunxp 4699 elrn2g 4835 eldmg 4840 elrnmpt 4894 elrnmpt1 4896 elimag 4992 elrnmpog 6010 eloprabi 6222 tfrlem3ag 6335 tfr1onlem3ag 6363 tfrcllemsucaccv 6380 elqsg 6612 elixp2 6729 isomni 7165 ismkv 7182 iswomni 7194 1idprl 7620 1idpru 7621 recexprlemell 7652 recexprlemelu 7653 mertenslemub 11577 mertenslemi1 11578 mertenslem2 11579 4sqexercise1 12433 4sqexercise2 12434 4sqlemsdc 12435 ismgm 12836 istopg 13976 isbasisg 14021 2sqlem8 14948 2sqlem9 14949 |
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