Theorem inass 3176

Description: Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)

Assertion

Ref	Expression
inass	|- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )

Proof of Theorem inass

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anass 393	. . . 4 |- ( ( ( x e. A /\ x e. B ) /\ x e. C ) <-> ( x e. A /\ ( x e. B /\ x e. C ) ) )
2		elin 3155	. . . . 5 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
3	2	anbi2i 444	. . . 4 |- ( ( x e. A /\ x e. ( B i^i C ) ) <-> ( x e. A /\ ( x e. B /\ x e. C ) ) )
4	1, 3	bitr4i 185	. . 3 |- ( ( ( x e. A /\ x e. B ) /\ x e. C ) <-> ( x e. A /\ x e. ( B i^i C ) ) )
5		elin 3155	. . . 4 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
6	5	anbi1i 445	. . 3 |- ( ( x e. ( A i^i B ) /\ x e. C ) <-> ( ( x e. A /\ x e. B ) /\ x e. C ) )
7		elin 3155	. . 3 |- ( x e. ( A i^i ( B i^i C ) ) <-> ( x e. A /\ x e. ( B i^i C ) ) )
8	4, 6, 7	3bitr4i 210	. 2 |- ( ( x e. ( A i^i B ) /\ x e. C ) <-> x e. ( A i^i ( B i^i C ) ) )
9	8	ineqri 3159	1 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   i^i cin 2972

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979

This theorem is referenced by:  in12  3177  in32  3178  in4  3182  indif2  3208  difun1  3224  dfrab3ss  3242  resres  4642  inres  4647  imainrect  4786