Theorem in12 1651

Description: A rearrangement of intersection.

Assertion

Ref	Expression
in12	⊢ (A ∩ (B ∩ C)) = (B ∩ (A ∩ C))

Proof of Theorem in12

Step	Hyp	Ref	Expression
1		incom 1636	. . 3 ⊢ (A ∩ B) = (B ∩ A)
2	1	ineq1i 1641	. 2 ⊢ ((A ∩ B) ∩ C) = ((B ∩ A) ∩ C)
3		inass 1650	. 2 ⊢ ((A ∩ B) ∩ C) = (A ∩ (B ∩ C))
4		inass 1650	. 2 ⊢ ((B ∩ A) ∩ C) = (B ∩ (A ∩ C))
5	2, 3, 4	3eqtr3 1124	1 ⊢ (A ∩ (B ∩ C)) = (B ∩ (A ∩ C))

Syntax hints:   = wceq 1091   ∩ cin 1486

This theorem is referenced by:  in4 1653  kmlem11 3590  fh1 5518  fh2 5519

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491

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