Theorem elabg 2918

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 2347	. 2 |- F/_ x A
2		nfv 1550	. 2 |- F/ x ps
3		elabg.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabgf 2914	1 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   e. wcel 2175   {cab 2190

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773

This theorem is referenced by:  elab2g  2919  intmin3  3911  finds  4647  elxpi  4690  elabrexg  5826  ovelrn  6094  elfi  7072  indpi  7454  peano5nnnn  8004  peano5nni  9038  lss1d  14116  lspsn  14149  zndvds  14382  eltg  14495  eltg2  14496