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Theorem elab2g 2750
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 2149 . 2 (𝐴𝐵𝐴 ∈ {𝑥𝜑})
3 elab2g.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
43elabg 2749 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
52, 4syl5bb 190 1 (𝐴𝑉 → (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  {cab 2069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614
This theorem is referenced by:  elab2  2751  elab4g  2752  eldif  2993  elun  3125  elin  3167  elsng  3437  elprg  3442  eluni  3630  eliun  3708  eliin  3709  elopab  4049  elong  4164  opeliunxp  4451  elrn2g  4584  eldmg  4589  elrnmpt  4642  elrnmpt1  4644  elimag  4733  elrnmpt2g  5692  eloprabi  5901  tfrlem3ag  6006  tfr1onlem3ag  6034  tfrcllemsucaccv  6051  elqsg  6272  isomni  6697  1idprl  7052  1idpru  7053  recexprlemell  7084  recexprlemelu  7085
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