ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elabgf Structured version   Unicode version

Theorem elabgf 2679
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabgf.1  F/_
elabgf.2  F/
elabgf.3
Assertion
Ref Expression
elabgf  {  |  }

Proof of Theorem elabgf
StepHypRef Expression
1 elabgf.1 . 2  F/_
2 nfab1 2177 . . . 4  F/_ {  |  }
31, 2nfel 2183 . . 3  F/  {  |  }
4 elabgf.2 . . 3  F/
53, 4nfbi 1478 . 2  F/  {  |  }
6 eleq1 2097 . . 3  {  |  }  {  |  }
7 elabgf.3 . . 3
86, 7bibi12d 224 . 2  {  |  }  {  |  }
9 abid 2025 . 2  {  |  }
101, 5, 8, 9vtoclgf 2606 1  {  |  }
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1242   F/wnf 1346   wcel 1390   {cab 2023   F/_wnfc 2162
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  elabf  2680  elabg  2682  elab3gf  2686  elrabf  2690  bj-intabssel  9263
  Copyright terms: Public domain W3C validator