Theorem elab2 2860

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 2859	. 2 |- ( A e. _V -> ( A e. B <-> ps ) )
5	1, 4	ax-mp 5	1 |- ( A e. B <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1335   e. wcel 2128   {cab 2143   _Vcvv 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714

This theorem is referenced by:  elpw  3549  elint  3813  opabid  4217  elrn2  4827  elimasn  4952  oprabid  5850  tfrlem3a  6254  tfrcllemsucaccv  6298  tfrcllembxssdm  6300  tfrcllemres  6306  addnqprlemrl  7471  addnqprlemru  7472  addnqprlemfl  7473  addnqprlemfu  7474  mulnqprlemrl  7487  mulnqprlemru  7488  mulnqprlemfl  7489  mulnqprlemfu  7490  ltnqpr  7507  ltnqpri  7508  archpr  7557  cauappcvgprlemladdfu  7568  cauappcvgprlemladdfl  7569  caucvgprlemladdfu  7591  caucvgprprlemopu  7613  suplocexprlemloc  7635  txuni2  12627