Theorem elab 2964

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 1577	. 2 |- F/ x ps
2		elab.1	. 2 |- A e. _V
3		elab.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabf 2963	1 |- ( A e. { x | ph } <-> 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:  ralab  2980  rexab  2982  intab  3984  dfiin2g  4030  dfiunv2  4033  uniuni  4578  dcextest  4709  peano5  4726  finds  4728  finds2  4729  funcnvuni  5431  fun11iun  5641  elabrex  5937  abrexco  5939  mapval2  6926  ssenen  7119  snexxph  7234  sbthlem2  7242  indpi  7674  nqprm  7874  nqprrnd  7875  nqprdisj  7876  nqprloc  7877  nqprl  7883  nqpru  7884  cauappcvgprlem2  7992  caucvgprlem2  8012  peano1nnnn  8184  peano2nnnn  8185  1nn  9269  peano2nn  9270  dfuzi  9710  hashfacen  11237  shftfvalg  11532  ovshftex  11533  shftfval  11535  4sqlemafi  13123  lss1d  14662  txdis1cn  15274  ushgredgedg  16352  ushgredgedgloop  16354  bj-ssom  16847