Theorem elab4g 2837

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 2700	. 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 2835	. 2 |- ( A e. _V -> ( A e. B <-> ps ) )
5	1, 4	biadan2 452	1 |- ( A e. B <-> ( A e. _V /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   {cab 2126   _Vcvv 2689

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691

This theorem is referenced by: (None)