Theorem elab 2883

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 1528	. 2 |- F/ x ps
2		elab.1	. 2 |- A e. _V
3		elab.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabf 2882	1 |- ( A e. { x | ph } <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   {cab 2163   _Vcvv 2739

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741

This theorem is referenced by:  ralab  2899  rexab  2901  intab  3875  dfiin2g  3921  dfiunv2  3924  uniuni  4453  dcextest  4582  peano5  4599  finds  4601  finds2  4602  funcnvuni  5287  fun11iun  5484  elabrex  5760  abrexco  5762  mapval2  6680  ssenen  6853  snexxph  6951  sbthlem2  6959  indpi  7343  nqprm  7543  nqprrnd  7544  nqprdisj  7545  nqprloc  7546  nqprl  7552  nqpru  7553  cauappcvgprlem2  7661  caucvgprlem2  7681  peano1nnnn  7853  peano2nnnn  7854  1nn  8932  peano2nn  8933  dfuzi  9365  hashfacen  10818  shftfvalg  10829  ovshftex  10830  shftfval  10832  lss1d  13475  txdis1cn  13817  bj-ssom  14727