Theorem elab2 2974

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 2973	. 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 1402   e. wcel 2209   {cab 2224   _Vcvv 2821

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  elpw  3691  elint  3971  opabid  4393  elrn2  5019  elimasn  5149  oprabid  6107  tfrlem3a  6571  tfrcllemsucaccv  6615  tfrcllembxssdm  6617  tfrcllemres  6623  addnqprlemrl  7914  addnqprlemru  7915  addnqprlemfl  7916  addnqprlemfu  7917  mulnqprlemrl  7930  mulnqprlemru  7931  mulnqprlemfl  7932  mulnqprlemfu  7933  ltnqpr  7950  ltnqpri  7951  archpr  8000  cauappcvgprlemladdfu  8011  cauappcvgprlemladdfl  8012  caucvgprlemladdfu  8034  caucvgprprlemopu  8056  suplocexprlemloc  8078  4sqlem12  13159  txuni2  15280