Theorem elabg 2972

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 2392	. 2 |- F/_ x A
2		nfv 1581	. 2 |- F/ x ps
3		elabg.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabgf 2968	1 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is used by:  elab2g  2973  intmin3  3997  finds  4749  elxpi  4792  elabrexg  5967  ovelrn  6241  elabreximd  6360  elfi  7310  indpi  7715  peano5nnnn  8265  peano5nni  9316  hashf1lem1  11307  lss1d  14829  lspsn  14862  zndvds  15093  eltg  15278  eltg2  15279  ausgrusgrien  16617