Theorem elab2 2925

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 2924	. 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 1373   e. wcel 2177   {cab 2192   _Vcvv 2773

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775

This theorem is referenced by:  elpw  3627  elint  3897  opabid  4310  elrn2  4929  elimasn  5058  oprabid  5989  tfrlem3a  6409  tfrcllemsucaccv  6453  tfrcllembxssdm  6455  tfrcllemres  6461  addnqprlemrl  7690  addnqprlemru  7691  addnqprlemfl  7692  addnqprlemfu  7693  mulnqprlemrl  7706  mulnqprlemru  7707  mulnqprlemfl  7708  mulnqprlemfu  7709  ltnqpr  7726  ltnqpri  7727  archpr  7776  cauappcvgprlemladdfu  7787  cauappcvgprlemladdfl  7788  caucvgprlemladdfu  7810  caucvgprprlemopu  7832  suplocexprlemloc  7854  4sqlem12  12800  txuni2  14803