Theorem eliin 3876

Description: Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)

Assertion

Ref	Expression
eliin	|- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   C( x)   V( x)

Proof of Theorem eliin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2233	. . 3 |- ( y = A -> ( y e. C <-> A e. C ) )
2	1	ralbidv 2470	. 2 |- ( y = A -> ( A. x e. B y e. C <-> A. x e. B A e. C ) )
3		df-iin 3874	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
4	2, 3	elab2g 2877	1 |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   |^|_ciin 3872

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-v 2732  df-iin 3874

This theorem is referenced by:  iinconstm  3880  iuniin  3881  iinss1  3883  ssiinf  3920  iinss  3922  iinss2  3923  iinab  3932  iundif2ss  3936  iindif2m  3938  iinin2m  3939  elriin  3941  iinpw  3961  xpiindim  4746  cnviinm  5150  iinerm  6581  ixpiinm  6698