Theorem elab2 2968

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 2967	. 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 1398   e. wcel 2205   {cab 2220   _Vcvv 2815

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817

This theorem is referenced by:  elpw  3680  elint  3960  opabid  4379  elrn2  5004  elimasn  5134  oprabid  6090  tfrlem3a  6554  tfrcllemsucaccv  6598  tfrcllembxssdm  6600  tfrcllemres  6606  addnqprlemrl  7888  addnqprlemru  7889  addnqprlemfl  7890  addnqprlemfu  7891  mulnqprlemrl  7904  mulnqprlemru  7905  mulnqprlemfl  7906  mulnqprlemfu  7907  ltnqpr  7924  ltnqpri  7925  archpr  7974  cauappcvgprlemladdfu  7985  cauappcvgprlemladdfl  7986  caucvgprlemladdfu  8008  caucvgprprlemopu  8030  suplocexprlemloc  8052  4sqlem12  13125  txuni2  15247