Theorem elab 2908

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 1542	. 2 |- F/ x ps
2		elab.1	. 2 |- A e. _V
3		elab.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabf 2907	1 |- ( A e. { x | ph } <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   {cab 2182   _Vcvv 2763

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  ralab  2924  rexab  2926  intab  3904  dfiin2g  3950  dfiunv2  3953  uniuni  4487  dcextest  4618  peano5  4635  finds  4637  finds2  4638  funcnvuni  5328  fun11iun  5526  elabrex  5805  abrexco  5807  mapval2  6738  ssenen  6913  snexxph  7017  sbthlem2  7025  indpi  7411  nqprm  7611  nqprrnd  7612  nqprdisj  7613  nqprloc  7614  nqprl  7620  nqpru  7621  cauappcvgprlem2  7729  caucvgprlem2  7749  peano1nnnn  7921  peano2nnnn  7922  1nn  9003  peano2nn  9004  dfuzi  9438  hashfacen  10930  shftfvalg  10985  ovshftex  10986  shftfval  10988  4sqlemafi  12574  lss1d  13949  txdis1cn  14524  bj-ssom  15592