Theorem elab 2739

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 1462	. 2 |- F/ x ps
2		elab.1	. 2 |- A e. _V
3		elab.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabf 2738	1 |- ( A e. { x | ph } <-> ps )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   e. wcel 1434   {cab 2068   _Vcvv 2602

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604

This theorem is referenced by:  ralab  2753  rexab  2755  intab  3673  dfiin2g  3719  dfiunv2  3722  uniuni  4209  peano5  4347  finds  4349  finds2  4350  funcnvuni  4999  fun11iun  5178  elabrex  5429  abrexco  5430  indpi  6594  nqprm  6794  nqprrnd  6795  nqprdisj  6796  nqprloc  6797  nqprl  6803  nqpru  6804  cauappcvgprlem2  6912  caucvgprlem2  6932  peano1nnnn  7082  peano2nnnn  7083  1nn  8117  peano2nn  8118  dfuzi  8538  shftfvalg  9844  ovshftex  9845  shftfval  9847  bj-ssom  10889