Theorem eliin 2561

Description: Membership in indexed intersection.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem eliin

Step	Hyp	Ref	Expression
1		eleq1 1526	. . 3 |- (y = A -> (y e. C <-> A e. C))
2	1	ralbidv 1655	. 2 |- (y = A -> (A.x e. B y e. C <-> A.x e. B A e. C))
3		df-iin 2559	. 2 |- |^|_x e. B C = {y | A.x e. B y e. C}
4	2, 3	elab2g 1891	1 |- (A e. D -> (A e. |^|_x e. B C <-> A.x e. B A e. C))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  A.wral 1637  |^|_ciin 2557

This theorem is referenced by:  iuniin 2563  ssiin 2589  iinss 2590  iinun2 2599  iundif2 2600  iindif2 2601  iinuni 2605  iinpw 2607  iintlem1 10476  iint 10478

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-iin 2559

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