Theorem elab 2832

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)

Hypotheses

Ref	Expression
elab.1	|- A e. _V
elab.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
elab	|- ( A e. { x | ph } <-> ps )

Distinct variable groups:   ps, x   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem elab

Step	Hyp	Ref	Expression
1		nfv 1509	. 2 |- F/ x ps
2		elab.1	. 2 |- A e. _V
3		elab.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabf 2831	1 |- ( A e. { x | ph } <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481   {cab 2126   _Vcvv 2689

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691

This theorem is referenced by:  ralab  2848  rexab  2850  intab  3808  dfiin2g  3854  dfiunv2  3857  uniuni  4380  dcextest  4503  peano5  4520  finds  4522  finds2  4523  funcnvuni  5200  fun11iun  5396  elabrex  5667  abrexco  5668  mapval2  6580  ssenen  6753  snexxph  6846  sbthlem2  6854  indpi  7174  nqprm  7374  nqprrnd  7375  nqprdisj  7376  nqprloc  7377  nqprl  7383  nqpru  7384  cauappcvgprlem2  7492  caucvgprlem2  7512  peano1nnnn  7684  peano2nnnn  7685  1nn  8755  peano2nn  8756  dfuzi  9185  hashfacen  10611  shftfvalg  10622  ovshftex  10623  shftfval  10625  txdis1cn  12486  bj-ssom  13305