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Theorem elab2g 2686
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 2104 . 2 (𝐴𝐵𝐴 ∈ {𝑥𝜑})
3 elab2g.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
43elabg 2685 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
52, 4syl5bb 181 1 (𝐴𝑉 → (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wcel 1393  {cab 2026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556
This theorem is referenced by:  elab2  2687  elab4g  2688  eldif  2924  elun  3081  elin  3123  elsng  3387  elprg  3392  eluni  3580  eliun  3658  eliin  3659  elopab  3992  elong  4082  opeliunxp  4357  elrn2g  4487  eldmg  4492  elrnmpt  4545  elrnmpt1  4547  elimag  4634  elrnmpt2g  5575  eloprabi  5784  tfrlem3ag  5886  elqsg  6117  1idprl  6643  1idpru  6644  recexprlemell  6675  recexprlemelu  6676
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