Theorem elab2 2965

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 2964	. 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 1398   e. wcel 2203   {cab 2218   _Vcvv 2813

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815

This theorem is referenced by:  elpw  3675  elint  3955  opabid  4374  elrn2  4999  elimasn  5129  oprabid  6082  tfrlem3a  6541  tfrcllemsucaccv  6585  tfrcllembxssdm  6587  tfrcllemres  6593  addnqprlemrl  7872  addnqprlemru  7873  addnqprlemfl  7874  addnqprlemfu  7875  mulnqprlemrl  7888  mulnqprlemru  7889  mulnqprlemfl  7890  mulnqprlemfu  7891  ltnqpr  7908  ltnqpri  7909  archpr  7958  cauappcvgprlemladdfu  7969  cauappcvgprlemladdfl  7970  caucvgprlemladdfu  7992  caucvgprprlemopu  8014  suplocexprlemloc  8036  4sqlem12  13100  txuni2  15121