Theorem elab 2828

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 1508	. 2 |- F/ x ps
2		elab.1	. 2 |- A e. _V
3		elab.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabf 2827	1 |- ( A e. { x | ph } <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2125   _Vcvv 2686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  ralab  2844  rexab  2846  intab  3800  dfiin2g  3846  dfiunv2  3849  uniuni  4372  dcextest  4495  peano5  4512  finds  4514  finds2  4515  funcnvuni  5192  fun11iun  5388  elabrex  5659  abrexco  5660  mapval2  6572  ssenen  6745  snexxph  6838  sbthlem2  6846  indpi  7150  nqprm  7350  nqprrnd  7351  nqprdisj  7352  nqprloc  7353  nqprl  7359  nqpru  7360  cauappcvgprlem2  7468  caucvgprlem2  7488  peano1nnnn  7660  peano2nnnn  7661  1nn  8731  peano2nn  8732  dfuzi  9161  hashfacen  10579  shftfvalg  10590  ovshftex  10591  shftfval  10593  txdis1cn  12447  bj-ssom  13134