Theorem wddi4 1110

Description: The weak distributive law in WDOL. (Contributed by NM, 5-Mar-2006.)

Assertion

Ref	Expression
wddi4	(((a ∩ b) ∪ c) ≡ ((a ∪ c) ∩ (b ∪ c))) = 1

Proof of Theorem wddi4

Step	Hyp	Ref	Expression
1		wa2 192	. 2 (((a ∩ b) ∪ c) ≡ (c ∪ (a ∩ b))) = 1
2		wddi3 1109	. . 3 ((c ∪ (a ∩ b)) ≡ ((c ∪ a) ∩ (c ∪ b))) = 1
3		wa2 192	. . . 4 ((c ∪ a) ≡ (a ∪ c)) = 1
4		wa2 192	. . . 4 ((c ∪ b) ≡ (b ∪ c)) = 1
5	3, 4	w2an 373	. . 3 (((c ∪ a) ∩ (c ∪ b)) ≡ ((a ∪ c) ∩ (b ∪ c))) = 1
6	2, 5	wr2 371	. 2 ((c ∪ (a ∩ b)) ≡ ((a ∪ c) ∩ (b ∪ c))) = 1
7	1, 6	wr2 371	1 (((a ∩ b) ∪ c) ≡ ((a ∪ c) ∩ (b ∪ c))) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8

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-wom 361  ax-wdol 1104

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134

This theorem is referenced by:  wdid0id5  1111