Theorem elab2 2912

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 2911	. 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 1364   e. wcel 2167   {cab 2182   _Vcvv 2763

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  elpw  3612  elint  3881  opabid  4291  elrn2  4909  elimasn  5037  oprabid  5955  tfrlem3a  6369  tfrcllemsucaccv  6413  tfrcllembxssdm  6415  tfrcllemres  6421  addnqprlemrl  7626  addnqprlemru  7627  addnqprlemfl  7628  addnqprlemfu  7629  mulnqprlemrl  7642  mulnqprlemru  7643  mulnqprlemfl  7644  mulnqprlemfu  7645  ltnqpr  7662  ltnqpri  7663  archpr  7712  cauappcvgprlemladdfu  7723  cauappcvgprlemladdfl  7724  caucvgprlemladdfu  7746  caucvgprprlemopu  7768  suplocexprlemloc  7790  4sqlem12  12581  txuni2  14502