Theorem elab2 2955

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 2954	. 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 1398   e. wcel 2202   {cab 2217   _Vcvv 2803

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805

This theorem is referenced by:  elpw  3662  elint  3939  opabid  4356  elrn2  4980  elimasn  5110  oprabid  6060  tfrlem3a  6519  tfrcllemsucaccv  6563  tfrcllembxssdm  6565  tfrcllemres  6571  addnqprlemrl  7820  addnqprlemru  7821  addnqprlemfl  7822  addnqprlemfu  7823  mulnqprlemrl  7836  mulnqprlemru  7837  mulnqprlemfl  7838  mulnqprlemfu  7839  ltnqpr  7856  ltnqpri  7857  archpr  7906  cauappcvgprlemladdfu  7917  cauappcvgprlemladdfl  7918  caucvgprlemladdfu  7940  caucvgprprlemopu  7962  suplocexprlemloc  7984  4sqlem12  13038  txuni2  15050