Theorem elab4g 2742

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)

Hypotheses

Ref	Expression
elab4g.1	|- ( x = A -> ( ph <-> ps ) )
elab4g.2	|- B = { x | ph }

Assertion

Ref	Expression
elab4g	|- ( A e. B <-> ( A e. _V /\ ps ) )

Distinct variable groups:   ps, x   x, A

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

Proof of Theorem elab4g

Step	Hyp	Ref	Expression
1		elex 2610	. 2 |- ( A e. B -> A e. _V )
2		elab4g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3		elab4g.2	. . 3 |- B = { x | ph }
4	2, 3	elab2g 2740	. 2 |- ( A e. _V -> ( A e. B <-> ps ) )
5	1, 4	biadan2 443	1 |- ( A e. B <-> ( A e. _V /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by: (None)