Theorem iineq2 4008

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 2296	. . . . 5 |- ( B = C -> ( y e. B <-> y e. C ) )
2	1	ralimi 2605	. . . 4 |- ( A. x e. A B = C -> A. x e. A ( y e. B <-> y e. C ) )
3		ralbi 2675	. . . 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 2352	. 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 3994	. 2 |- |^|_ x e. A B = { y | A. x e. A y e. B }
7		df-iin 3994	. 2 |- |^|_ x e. A C = { y | A. x e. A y e. C }
8	5, 6, 7	3eqtr4g 2290	1 |- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   |^|_ciin 3992

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-ral 2525  df-iin 3994

This theorem is referenced by:  iineq2i  4010  iineq2d  4011