Theorem elab2 2952

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 2951	. 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 1395   e. wcel 2200   {cab 2215   _Vcvv 2800

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802

This theorem is referenced by:  elpw  3656  elint  3932  opabid  4348  elrn2  4972  elimasn  5101  oprabid  6045  tfrlem3a  6471  tfrcllemsucaccv  6515  tfrcllembxssdm  6517  tfrcllemres  6523  addnqprlemrl  7767  addnqprlemru  7768  addnqprlemfl  7769  addnqprlemfu  7770  mulnqprlemrl  7783  mulnqprlemru  7784  mulnqprlemfl  7785  mulnqprlemfu  7786  ltnqpr  7803  ltnqpri  7804  archpr  7853  cauappcvgprlemladdfu  7864  cauappcvgprlemladdfl  7865  caucvgprlemladdfu  7887  caucvgprprlemopu  7909  suplocexprlemloc  7931  4sqlem12  12965  txuni2  14970