Theorem elabg 2876

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 2312	. 2 |- F/_ x A
2		nfv 1521	. 2 |- F/ x ps
3		elabg.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabgf 2872	1 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  elab2g  2877  intmin3  3858  finds  4584  elxpi  4627  ovelrn  6001  elfi  6948  indpi  7304  peano5nnnn  7854  peano5nni  8881  eltg  12846  eltg2  12847