Theorem elintrab 3648

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)

Hypothesis

Ref	Expression
inteqab.1	|- A e. _V

Assertion

Ref	Expression
elintrab	|- ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) )

Distinct variable group:   x, A

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

Proof of Theorem elintrab

Step	Hyp	Ref	Expression
1		inteqab.1	. . . 4 |- A e. _V
2	1	elintab 3647	. . 3 |- ( A e. |^| { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. B /\ ph ) -> A e. x ) )
3		impexp 259	. . . 4 |- ( ( ( x e. B /\ ph ) -> A e. x ) <-> ( x e. B -> ( ph -> A e. x ) ) )
4	3	albii 1399	. . 3 |- ( A. x ( ( x e. B /\ ph ) -> A e. x ) <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
5	2, 4	bitri 182	. 2 |- ( A e. |^| { x | ( x e. B /\ ph ) } <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
6		df-rab 2357	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
7	6	inteqi 3640	. . 3 |- |^| { x e. B | ph } = |^| { x | ( x e. B /\ ph ) }
8	7	eleq2i 2145	. 2 |- ( A e. |^| { x e. B | ph } <-> A e. |^| { x | ( x e. B /\ ph ) } )
9		df-ral 2353	. 2 |- ( A. x e. B ( ph -> A e. x ) <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
10	5, 8, 9	3bitr4i 210	1 |- ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   e. wcel 1433   {cab 2067   A.wral 2348   {crab 2352   _Vcvv 2601   |^|cint 3636

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-ral 2353  df-rab 2357  df-v 2603  df-int 3637

This theorem is referenced by:  elintrabg  3649  intmin  3656  bj-indint  10726