Theorem iineq2 3982

Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iineq2	|- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C )

Proof of Theorem iineq2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2293	. . . . 5 |- ( B = C -> ( y e. B <-> y e. C ) )
2	1	ralimi 2593	. . . 4 |- ( A. x e. A B = C -> A. x e. A ( y e. B <-> y e. C ) )
3		ralbi 2663	. . . 4 |- ( A. x e. A ( y e. B <-> y e. C ) -> ( A. x e. A y e. B <-> A. x e. A y e. C ) )
4	2, 3	syl 14	. . 3 |- ( A. x e. A B = C -> ( A. x e. A y e. B <-> A. x e. A y e. C ) )
5	4	abbidv 2347	. 2 |- ( A. x e. A B = C -> { y | A. x e. A y e. B } = { y | A. x e. A y e. C } )
6		df-iin 3968	. 2 |- |^|_ x e. A B = { y | A. x e. A y e. B }
7		df-iin 3968	. 2 |- |^|_ x e. A C = { y | A. x e. A y e. C }
8	5, 6, 7	3eqtr4g 2287	1 |- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   |^|_ciin 3966

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-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-iin 3968

This theorem is referenced by:  iineq2i  3984  iineq2d  3985