Theorem elabg 2950

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)

Hypothesis

Ref	Expression
elabg.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
elabg	|- ( A e. V -> ( A e. { x | ph } <-> ps ) )

Distinct variable groups:   ps, x   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem elabg

Step	Hyp	Ref	Expression
1		nfcv 2372	. 2 |- F/_ x A
2		nfv 1574	. 2 |- F/ x ps
3		elabg.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabgf 2946	1 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )

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

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 2802

This theorem is referenced by:  elab2g  2951  intmin3  3953  finds  4696  elxpi  4739  elabrexg  5894  ovelrn  6166  elfi  7161  indpi  7552  peano5nnnn  8102  peano5nni  9136  lss1d  14387  lspsn  14420  zndvds  14653  eltg  14766  eltg2  14767  ausgrusgrien  16010