Theorem eliin 3906

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 2252	. . 3 |- ( y = A -> ( y e. C <-> A e. C ) )
2	1	ralbidv 2490	. 2 |- ( y = A -> ( A. x e. B y e. C <-> A. x e. B A e. C ) )
3		df-iin 3904	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
4	2, 3	elab2g 2899	1 |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2160   A.wral 2468   |^|_ciin 3902

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-iin 3904

This theorem is referenced by:  iinconstm  3910  iuniin  3911  iinss1  3913  ssiinf  3951  iinss  3953  iinss2  3954  iinab  3963  iundif2ss  3967  iindif2m  3969  iinin2m  3970  elriin  3972  iinpw  3992  xpiindim  4782  cnviinm  5188  iinerm  6632  ixpiinm  6749