Theorem elab2 2785

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 2784	. 2 |- ( A e. _V -> ( A e. B <-> ps ) )
5	1, 4	ax-mp 7	1 |- ( A e. B <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1299   e. wcel 1448   {cab 2086   _Vcvv 2641

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643

This theorem is referenced by:  elpw  3463  elint  3724  opabid  4117  elrn2  4719  elimasn  4842  oprabid  5735  tfrlem3a  6137  tfrcllemsucaccv  6181  tfrcllembxssdm  6183  tfrcllemres  6189  addnqprlemrl  7266  addnqprlemru  7267  addnqprlemfl  7268  addnqprlemfu  7269  mulnqprlemrl  7282  mulnqprlemru  7283  mulnqprlemfl  7284  mulnqprlemfu  7285  ltnqpr  7302  ltnqpri  7303  archpr  7352  cauappcvgprlemladdfu  7363  cauappcvgprlemladdfl  7364  caucvgprlemladdfu  7386  caucvgprprlemopu  7408  txuni2  12206