Theorem elab2 2951

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 2950	. 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 1395   e. wcel 2200   {cab 2215   _Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  elpw  3655  elint  3928  opabid  4343  elrn2  4965  elimasn  5094  oprabid  6032  tfrlem3a  6454  tfrcllemsucaccv  6498  tfrcllembxssdm  6500  tfrcllemres  6506  addnqprlemrl  7740  addnqprlemru  7741  addnqprlemfl  7742  addnqprlemfu  7743  mulnqprlemrl  7756  mulnqprlemru  7757  mulnqprlemfl  7758  mulnqprlemfu  7759  ltnqpr  7776  ltnqpri  7777  archpr  7826  cauappcvgprlemladdfu  7837  cauappcvgprlemladdfl  7838  caucvgprlemladdfu  7860  caucvgprprlemopu  7882  suplocexprlemloc  7904  4sqlem12  12920  txuni2  14924