Theorem elab2 2754

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 2753	. 2 |- ( A e. _V -> ( A e. B <-> ps ) )
5	1, 4	ax-mp 7	1 |- ( A e. B <-> ps )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1287   e. wcel 1436   {cab 2071   _Vcvv 2615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617

This theorem is referenced by:  elpw  3421  elint  3679  opabid  4060  elrn2  4647  elimasn  4768  oprabid  5640  tfrlem3a  6031  tfrcllemsucaccv  6075  tfrcllembxssdm  6077  tfrcllemres  6083  addnqprlemrl  7063  addnqprlemru  7064  addnqprlemfl  7065  addnqprlemfu  7066  mulnqprlemrl  7079  mulnqprlemru  7080  mulnqprlemfl  7081  mulnqprlemfu  7082  ltnqpr  7099  ltnqpri  7100  archpr  7149  cauappcvgprlemladdfu  7160  cauappcvgprlemladdfl  7161  caucvgprlemladdfu  7183  caucvgprprlemopu  7205