Theorem elabg 2910

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 2339	. 2 |- F/_ x A
2		nfv 1542	. 2 |- F/ x ps
3		elabg.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabgf 2906	1 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   {cab 2182

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  elab2g  2911  intmin3  3902  finds  4637  elxpi  4680  elabrexg  5808  ovelrn  6076  elfi  7046  indpi  7426  peano5nnnn  7976  peano5nni  9010  lss1d  14015  lspsn  14048  zndvds  14281  eltg  14372  eltg2  14373