Theorem iineq2 3825

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 2201	. . . . 5 |- ( B = C -> ( y e. B <-> y e. C ) )
2	1	ralimi 2493	. . . 4 |- ( A. x e. A B = C -> A. x e. A ( y e. B <-> y e. C ) )
3		ralbi 2562	. . . 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 2255	. 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 3811	. 2 |- |^|_ x e. A B = { y | A. x e. A y e. B }
7		df-iin 3811	. 2 |- |^|_ x e. A C = { y | A. x e. A y e. C }
8	5, 6, 7	3eqtr4g 2195	1 |- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2123   A.wral 2414   |^|_ciin 3809

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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-iin 3811

This theorem is referenced by:  iineq2i  3827  iineq2d  3828