Theorem elab 2738

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 1461	. 2 |- F/ x ps
2		elab.1	. 2 |- A e. _V
3		elab.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabf 2737	1 |- ( A e. { x | ph } <-> ps )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  ralab  2752  rexab  2754  intab  3665  dfiin2g  3711  dfiunv2  3714  uniuni  4201  peano5  4339  finds  4341  finds2  4342  funcnvuni  4988  fun11iun  5167  elabrex  5418  abrexco  5419  indpi  6532  nqprm  6732  nqprrnd  6733  nqprdisj  6734  nqprloc  6735  nqprl  6741  nqpru  6742  cauappcvgprlem2  6850  caucvgprlem2  6870  peano1nnnn  7020  peano2nnnn  7021  1nn  8050  peano2nn  8051  dfuzi  8457  shftfvalg  9706  ovshftex  9707  shftfval  9709  bj-ssom  10731