Theorem elab2 2967

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 2966	. 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  3677  elint  3957  opabid  4376  elrn2  5001  elimasn  5131  oprabid  6084  tfrlem3a  6543  tfrcllemsucaccv  6587  tfrcllembxssdm  6589  tfrcllemres  6595  addnqprlemrl  7874  addnqprlemru  7875  addnqprlemfl  7876  addnqprlemfu  7877  mulnqprlemrl  7890  mulnqprlemru  7891  mulnqprlemfl  7892  mulnqprlemfu  7893  ltnqpr  7910  ltnqpri  7911  archpr  7960  cauappcvgprlemladdfu  7971  cauappcvgprlemladdfl  7972  caucvgprlemladdfu  7994  caucvgprprlemopu  8016  suplocexprlemloc  8038  4sqlem12  13104  txuni2  15138