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Theorem elab2g 2831
 Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
Hypotheses
Ref Expression
elab2g.1 (𝑥 = 𝐴 → (𝜑𝜓))
elab2g.2 𝐵 = {𝑥𝜑}
Assertion
Ref Expression
elab2g (𝐴𝑉 → (𝐴𝐵𝜓))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem elab2g
StepHypRef Expression
1 elab2g.2 . . 3 𝐵 = {𝑥𝜑}
21eleq2i 2206 . 2 (𝐴𝐵𝐴 ∈ {𝑥𝜑})
3 elab2g.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
43elabg 2830 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
52, 4syl5bb 191 1 (𝐴𝑉 → (𝐴𝐵𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2125 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 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688 This theorem is referenced by:  elab2  2832  elab4g  2833  eldif  3080  elun  3217  elin  3259  elsng  3542  elprg  3547  eluni  3739  eliun  3817  eliin  3818  elopab  4180  elong  4295  opeliunxp  4594  elrn2g  4729  eldmg  4734  elrnmpt  4788  elrnmpt1  4790  elimag  4885  elrnmpog  5883  eloprabi  6094  tfrlem3ag  6206  tfr1onlem3ag  6234  tfrcllemsucaccv  6251  elqsg  6479  elixp2  6596  isomni  7008  ismkv  7027  iswomni  7037  1idprl  7412  1idpru  7413  recexprlemell  7444  recexprlemelu  7445  mertenslemub  11317  mertenslemi1  11318  mertenslem2  11319  istopg  12182  isbasisg  12227
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