Theorem eliin 3996

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 2295	. . 3 |- ( y = A -> ( y e. C <-> A e. C ) )
2	1	ralbidv 2542	. 2 |- ( y = A -> ( A. x e. B y e. C <-> A. x e. B A e. C ) )
3		df-iin 3994	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
4	2, 3	elab2g 2964	1 |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   |^|_ciin 3992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-iin 3994

This theorem is referenced by:  iinconstm  4000  iuniin  4001  iinss1  4003  ssiinf  4041  iinss  4043  iinss2  4044  iinab  4053  iundif2ss  4057  iindif2m  4059  iinin2m  4060  elriin  4062  iinpw  4082  xpiindim  4892  cnviinm  5304  iinerm  6841  ixpiinm  6959