Theorem elintrab 3843

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 3842	. . 3 |- ( A e. |^| { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. B /\ ph ) -> A e. x ) )
3		impexp 261	. . . 4 |- ( ( ( x e. B /\ ph ) -> A e. x ) <-> ( x e. B -> ( ph -> A e. x ) ) )
4	3	albii 1463	. . 3 |- ( A. x ( ( x e. B /\ ph ) -> A e. x ) <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
5	2, 4	bitri 183	. 2 |- ( A e. |^| { x | ( x e. B /\ ph ) } <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
6		df-rab 2457	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
7	6	inteqi 3835	. . 3 |- |^| { x e. B | ph } = |^| { x | ( x e. B /\ ph ) }
8	7	eleq2i 2237	. 2 |- ( A e. |^| { x e. B | ph } <-> A e. |^| { x | ( x e. B /\ ph ) } )
9		df-ral 2453	. 2 |- ( A. x e. B ( ph -> A e. x ) <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
10	5, 8, 9	3bitr4i 211	1 |- ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   e. wcel 2141   {cab 2156   A.wral 2448   {crab 2452   _Vcvv 2730   |^|cint 3831

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-int 3832

This theorem is referenced by:  elintrabg  3844  intmin  3851  bj-indint  13966