Theorem elab2 2832

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 2831	. 2 |- ( A e. _V -> ( A e. B <-> ps ) )
5	1, 4	ax-mp 5	1 |- ( A e. B <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2125   _Vcvv 2686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  elpw  3516  elint  3777  opabid  4179  elrn2  4781  elimasn  4906  oprabid  5803  tfrlem3a  6207  tfrcllemsucaccv  6251  tfrcllembxssdm  6253  tfrcllemres  6259  addnqprlemrl  7365  addnqprlemru  7366  addnqprlemfl  7367  addnqprlemfu  7368  mulnqprlemrl  7381  mulnqprlemru  7382  mulnqprlemfl  7383  mulnqprlemfu  7384  ltnqpr  7401  ltnqpri  7402  archpr  7451  cauappcvgprlemladdfu  7462  cauappcvgprlemladdfl  7463  caucvgprlemladdfu  7485  caucvgprprlemopu  7507  suplocexprlemloc  7529  txuni2  12425