Theorem elab2 2900

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 2899	. 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 1364   e. wcel 2160   {cab 2175   _Vcvv 2752

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754

This theorem is referenced by:  elpw  3596  elint  3865  opabid  4275  elrn2  4887  elimasn  5013  oprabid  5928  tfrlem3a  6335  tfrcllemsucaccv  6379  tfrcllembxssdm  6381  tfrcllemres  6387  addnqprlemrl  7586  addnqprlemru  7587  addnqprlemfl  7588  addnqprlemfu  7589  mulnqprlemrl  7602  mulnqprlemru  7603  mulnqprlemfl  7604  mulnqprlemfu  7605  ltnqpr  7622  ltnqpri  7623  archpr  7672  cauappcvgprlemladdfu  7683  cauappcvgprlemladdfl  7684  caucvgprlemladdfu  7706  caucvgprprlemopu  7728  suplocexprlemloc  7750  4sqlem12  12434  txuni2  14216