Theorem iineq2 3830

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 2203	. . . . 5 |- ( B = C -> ( y e. B <-> y e. C ) )
2	1	ralimi 2495	. . . 4 |- ( A. x e. A B = C -> A. x e. A ( y e. B <-> y e. C ) )
3		ralbi 2564	. . . 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 2257	. 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 3816	. 2 |- |^|_ x e. A B = { y | A. x e. A y e. B }
7		df-iin 3816	. 2 |- |^|_ x e. A C = { y | A. x e. A y e. C }
8	5, 6, 7	3eqtr4g 2197	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 2125   A.wral 2416   |^|_ciin 3814

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 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-iin 3816

This theorem is referenced by:  iineq2i  3832  iineq2d  3833