Theorem u4lem2 747

Description: Lemma for unified implication study. (Contributed by NM, 24-Dec-1997.)

Assertion

Ref	Expression
u4lem2	(((a →4 b) →4 a) →4 a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Proof of Theorem u4lem2

Step	Hyp	Ref	Expression
1		u4lemc1 683	. . . 4 a C ((a →4 b) →4 a)
2	1	comcom 453	. . 3 ((a →4 b) →4 a) C a
3	2	u4lemc4 704	. 2 (((a →4 b) →4 a) →4 a) = (((a →4 b) →4 a)⊥ ∪ a)
4		u4lem1n 742	. . . 4 ((a →4 b) →4 a)⊥ = ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
5	4	ax-r5 38	. . 3 (((a →4 b) →4 a)⊥ ∪ a) = (((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a)
6		ax-a3 32	. . . 4 (((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a) = ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a))
7		lear 161	. . . . . . 7 (((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ≤ a
8		leor 159	. . . . . . 7 a ≤ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
9	7, 8	letr 137	. . . . . 6 (((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ≤ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
10	9	df-le2 131	. . . . 5 ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
11		ax-a2 31	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
12	10, 11	ax-r2 36	. . . 4 ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
13	6, 12	ax-r2 36	. . 3 (((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
14	5, 13	ax-r2 36	. 2 (((a →4 b) →4 a)⊥ ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
15	3, 14	ax-r2 36	1 (((a →4 b) →4 a) →4 a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 15

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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)