Theorem inass 3252

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 396	. . . 4 |- ( ( ( x e. A /\ x e. B ) /\ x e. C ) <-> ( x e. A /\ ( x e. B /\ x e. C ) ) )
2		elin 3225	. . . . 5 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
3	2	anbi2i 450	. . . 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 3225	. . . 4 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
6	5	anbi1i 451	. . 3 |- ( ( x e. ( A i^i B ) /\ x e. C ) <-> ( ( x e. A /\ x e. B ) /\ x e. C ) )
7		elin 3225	. . 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 3235	1 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )

Syntax hints:   /\ wa 103   = wceq 1314   e. wcel 1463   i^i cin 3036

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-in 3043

This theorem is referenced by:  in12  3253  in32  3254  in4  3258  indif2  3286  difun1  3302  dfrab3ss  3320  resres  4789  inres  4794  imainrect  4942  restco  12186  restopnb  12193