Theorem elab2 2742

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Hypotheses

Ref	Expression
elab2.1	|- A e. _V
elab2.2	|- ( x = A -> ( ph <-> ps ) )
elab2.3	|- B = { x | ph }

Assertion

Ref	Expression
elab2	|- ( A e. B <-> ps )

Distinct variable groups:   ps, x   x, A

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem elab2

Step	Hyp	Ref	Expression
1		elab2.1	. 2 |- A e. _V
2		elab2.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3		elab2.3	. . 3 |- B = { x | ph }
4	2, 3	elab2g 2741	. 2 |- ( A e. _V -> ( A e. B <-> ps ) )
5	1, 4	ax-mp 7	1 |- ( A e. B <-> ps )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   e. wcel 1434   {cab 2068   _Vcvv 2602

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 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604

This theorem is referenced by:  elpw  3396  elint  3650  opabid  4020  elrn2  4604  elimasn  4722  oprabid  5568  tfrlem3a  5959  tfrcllemsucaccv  6003  tfrcllembxssdm  6005  tfrcllemres  6011  addnqprlemrl  6809  addnqprlemru  6810  addnqprlemfl  6811  addnqprlemfu  6812  mulnqprlemrl  6825  mulnqprlemru  6826  mulnqprlemfl  6827  mulnqprlemfu  6828  ltnqpr  6845  ltnqpri  6846  archpr  6895  cauappcvgprlemladdfu  6906  cauappcvgprlemladdfl  6907  caucvgprlemladdfu  6929  caucvgprprlemopu  6951