Theorem elab2 2908

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 2907	. 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 2164   {cab 2179   _Vcvv 2760

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762

This theorem is referenced by:  elpw  3607  elint  3876  opabid  4286  elrn2  4904  elimasn  5032  oprabid  5950  tfrlem3a  6363  tfrcllemsucaccv  6407  tfrcllembxssdm  6409  tfrcllemres  6415  addnqprlemrl  7617  addnqprlemru  7618  addnqprlemfl  7619  addnqprlemfu  7620  mulnqprlemrl  7633  mulnqprlemru  7634  mulnqprlemfl  7635  mulnqprlemfu  7636  ltnqpr  7653  ltnqpri  7654  archpr  7703  cauappcvgprlemladdfu  7714  cauappcvgprlemladdfl  7715  caucvgprlemladdfu  7737  caucvgprprlemopu  7759  suplocexprlemloc  7781  4sqlem12  12540  txuni2  14424