Theorem inass 3332

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 399	. . . 4 |- ( ( ( x e. A /\ x e. B ) /\ x e. C ) <-> ( x e. A /\ ( x e. B /\ x e. C ) ) )
2		elin 3305	. . . . 5 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
3	2	anbi2i 453	. . . 4 |- ( ( x e. A /\ x e. ( B i^i C ) ) <-> ( x e. A /\ ( x e. B /\ x e. C ) ) )
4	1, 3	bitr4i 186	. . 3 |- ( ( ( x e. A /\ x e. B ) /\ x e. C ) <-> ( x e. A /\ x e. ( B i^i C ) ) )
5		elin 3305	. . . 4 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
6	5	anbi1i 454	. . 3 |- ( ( x e. ( A i^i B ) /\ x e. C ) <-> ( ( x e. A /\ x e. B ) /\ x e. C ) )
7		elin 3305	. . 3 |- ( x e. ( A i^i ( B i^i C ) ) <-> ( x e. A /\ x e. ( B i^i C ) ) )
8	4, 6, 7	3bitr4i 211	. 2 |- ( ( x e. ( A i^i B ) /\ x e. C ) <-> x e. ( A i^i ( B i^i C ) ) )
9	8	ineqri 3315	1 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   i^i cin 3115

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122

This theorem is referenced by:  in12  3333  in32  3334  in4  3338  indif2  3366  difun1  3382  dfrab3ss  3400  resres  4896  inres  4901  imainrect  5049  restco  12814  restopnb  12821