Theorem elab2 2920

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 2919	. 2 |- ( A e. _V -> ( A e. B <-> ps ) )
5	1, 4	ax-mp 5	1 |- ( A e. B <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   e. wcel 2175   {cab 2190   _Vcvv 2771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773

This theorem is referenced by:  elpw  3621  elint  3890  opabid  4301  elrn2  4919  elimasn  5048  oprabid  5975  tfrlem3a  6395  tfrcllemsucaccv  6439  tfrcllembxssdm  6441  tfrcllemres  6447  addnqprlemrl  7669  addnqprlemru  7670  addnqprlemfl  7671  addnqprlemfu  7672  mulnqprlemrl  7685  mulnqprlemru  7686  mulnqprlemfl  7687  mulnqprlemfu  7688  ltnqpr  7705  ltnqpri  7706  archpr  7755  cauappcvgprlemladdfu  7766  cauappcvgprlemladdfl  7767  caucvgprlemladdfu  7789  caucvgprprlemopu  7811  suplocexprlemloc  7833  4sqlem12  12667  txuni2  14670