Theorem anandir 107

Description: Distribution of conjunction over conjunction.

Assertion

Ref	Expression
anandir	((a ∩ b) ∩ c) = ((a ∩ c) ∩ (b ∩ c))

Proof of Theorem anandir

Step	Hyp	Ref	Expression
1		anidm 103	. . . 4 (c ∩ c) = c
2	1	ax-r1 34	. . 3 c = (c ∩ c)
3	2	lan 70	. 2 ((a ∩ b) ∩ c) = ((a ∩ b) ∩ (c ∩ c))
4		an4 78	. 2 ((a ∩ b) ∩ (c ∩ c)) = ((a ∩ c) ∩ (b ∩ c))
5	3, 4	ax-r2 35	1 ((a ∩ b) ∩ c) = ((a ∩ c) ∩ (b ∩ c))

Syntax hints:   = wb 1   ∩ wa 7

This theorem is referenced by:  leran 145  ka4lemo 220  wr5-2v 348  wleran 376  ska4 415  i3orlem5 538  ud2lem1 545  mlaoml 815  comanblem2 853  oath1 984  4oath1 1020

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41

metamath.org