Theorem inass 3374

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 401	. . . 4 |- ( ( ( x e. A /\ x e. B ) /\ x e. C ) <-> ( x e. A /\ ( x e. B /\ x e. C ) ) )
2		elin 3347	. . . . 5 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
3	2	anbi2i 457	. . . 4 |- ( ( x e. A /\ x e. ( B i^i C ) ) <-> ( x e. A /\ ( x e. B /\ x e. C ) ) )
4	1, 3	bitr4i 187	. . 3 |- ( ( ( x e. A /\ x e. B ) /\ x e. C ) <-> ( x e. A /\ x e. ( B i^i C ) ) )
5		elin 3347	. . . 4 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
6	5	anbi1i 458	. . 3 |- ( ( x e. ( A i^i B ) /\ x e. C ) <-> ( ( x e. A /\ x e. B ) /\ x e. C ) )
7		elin 3347	. . 3 |- ( x e. ( A i^i ( B i^i C ) ) <-> ( x e. A /\ x e. ( B i^i C ) ) )
8	4, 6, 7	3bitr4i 212	. 2 |- ( ( x e. ( A i^i B ) /\ x e. C ) <-> x e. ( A i^i ( B i^i C ) ) )
9	8	ineqri 3357	1 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2167   i^i cin 3156

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163

This theorem is referenced by:  in12  3375  in32  3376  in4  3380  indif2  3408  difun1  3424  dfrab3ss  3442  resres  4959  inres  4964  imainrect  5116  ressinbasd  12777  ressressg  12778  restco  14494  restopnb  14501