Theorem distid 887

Description: Distributive law for identity. (Contributed by NM, 17-Mar-2002.)

Assertion

Ref	Expression
distid	((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) = (((a ≡ b) ∩ (a ≡ c)) ∪ ((a ≡ b) ∩ (b ≡ c)))

Proof of Theorem distid

Step	Hyp	Ref	Expression
1		lea 160	. . . 4 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (a ≡ b)
2		mlaconjo 886	. . . 4 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (a ≡ c)
3	1, 2	ler2an 173	. . 3 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ ((a ≡ b) ∩ (a ≡ c))
4		bicom 96	. . . . . 6 (a ≡ b) = (b ≡ a)
5		ax-a2 31	. . . . . 6 ((a ≡ c) ∪ (b ≡ c)) = ((b ≡ c) ∪ (a ≡ c))
6	4, 5	2an 79	. . . . 5 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) = ((b ≡ a) ∩ ((b ≡ c) ∪ (a ≡ c)))
7		mlaconjo 886	. . . . 5 ((b ≡ a) ∩ ((b ≡ c) ∪ (a ≡ c))) ≤ (b ≡ c)
8	6, 7	bltr 138	. . . 4 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (b ≡ c)
9	1, 8	ler2an 173	. . 3 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ ((a ≡ b) ∩ (b ≡ c))
10	3, 9	ler2or 172	. 2 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (((a ≡ b) ∩ (a ≡ c)) ∪ ((a ≡ b) ∩ (b ≡ c)))
11		ledi 174	. 2 (((a ≡ b) ∩ (a ≡ c)) ∪ ((a ≡ b) ∩ (b ≡ c))) ≤ ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c)))
12	10, 11	lebi 145	1 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) = (((a ≡ b) ∩ (a ≡ c)) ∪ ((a ≡ b) ∩ (b ≡ c)))

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6   ∩ wa 7

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)