Theorem elab 2947

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 1574	. 2 |- F/ x ps
2		elab.1	. 2 |- A e. _V
3		elab.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabf 2946	1 |- ( A e. { x | ph } <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   _Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  ralab  2963  rexab  2965  intab  3951  dfiin2g  3997  dfiunv2  4000  uniuni  4539  dcextest  4670  peano5  4687  finds  4689  finds2  4690  funcnvuni  5386  fun11iun  5589  elabrex  5874  abrexco  5876  mapval2  6815  ssenen  7000  snexxph  7105  sbthlem2  7113  indpi  7517  nqprm  7717  nqprrnd  7718  nqprdisj  7719  nqprloc  7720  nqprl  7726  nqpru  7727  cauappcvgprlem2  7835  caucvgprlem2  7855  peano1nnnn  8027  peano2nnnn  8028  1nn  9109  peano2nn  9110  dfuzi  9545  hashfacen  11045  shftfvalg  11315  ovshftex  11316  shftfval  11318  4sqlemafi  12904  lss1d  14332  txdis1cn  14937  ushgredgedg  16009  ushgredgedgloop  16011  bj-ssom  16229