Theorem distid 869

Description: Distributive law for identity.

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 152	. . . 4 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (a ≡ b)
2		mlaconjo 868	. . . 4 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (a ≡ c)
3	1, 2	ler2an 165	. . 3 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ ((a ≡ b) ∩ (a ≡ c))
4		bicom 88	. . . . . 6 (a ≡ b) = (b ≡ a)
5		ax-a2 30	. . . . . 6 ((a ≡ c) ∪ (b ≡ c)) = ((b ≡ c) ∪ (a ≡ c))
6	4, 5	2an 72	. . . . 5 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) = ((b ≡ a) ∩ ((b ≡ c) ∪ (a ≡ c)))
7		mlaconjo 868	. . . . 5 ((b ≡ a) ∩ ((b ≡ c) ∪ (a ≡ c))) ≤ (b ≡ c)
8	6, 7	bltr 130	. . . 4 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (b ≡ c)
9	1, 8	ler2an 165	. . 3 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ ((a ≡ b) ∩ (b ≡ c))
10	3, 9	ler2or 164	. 2 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (((a ≡ b) ∩ (a ≡ c)) ∪ ((a ≡ b) ∩ (b ≡ c)))
11		ledi 166	. 2 (((a ≡ b) ∩ (a ≡ c)) ∪ ((a ≡ b) ∩ (b ≡ c))) ≤ ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c)))
12	10, 11	lebi 137	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 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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