Theorem elab2 2951

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 2950	. 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 2799

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 2801

This theorem is referenced by:  elpw  3655  elint  3929  opabid  4344  elrn2  4966  elimasn  5095  oprabid  6039  tfrlem3a  6462  tfrcllemsucaccv  6506  tfrcllembxssdm  6508  tfrcllemres  6514  addnqprlemrl  7755  addnqprlemru  7756  addnqprlemfl  7757  addnqprlemfu  7758  mulnqprlemrl  7771  mulnqprlemru  7772  mulnqprlemfl  7773  mulnqprlemfu  7774  ltnqpr  7791  ltnqpri  7792  archpr  7841  cauappcvgprlemladdfu  7852  cauappcvgprlemladdfl  7853  caucvgprlemladdfu  7875  caucvgprprlemopu  7897  suplocexprlemloc  7919  4sqlem12  12940  txuni2  14945