Theorem u4lem3 752

Description: Lemma for unified implication study. (Contributed by NM, 17-Dec-1997.)

Assertion

Ref	Expression
u4lem3	(a →4 (b →4 a)) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))

Proof of Theorem u4lem3

Step	Hyp	Ref	Expression
1		u4lemc1 683	. . 3 a C (b →4 a)
2	1	u4lemc4 704	. 2 (a →4 (b →4 a)) = (a⊥ ∪ (b →4 a))
3		ax-a2 31	. . 3 (a⊥ ∪ (b →4 a)) = ((b →4 a) ∪ a⊥ )
4		u4lemonb 638	. . . 4 ((b →4 a) ∪ a⊥ ) = (((b ∩ a) ∪ (b⊥ ∩ a)) ∪ a⊥ )
5		ancom 74	. . . . . . 7 (b ∩ a) = (a ∩ b)
6		ancom 74	. . . . . . 7 (b⊥ ∩ a) = (a ∩ b⊥ )
7	5, 6	2or 72	. . . . . 6 ((b ∩ a) ∪ (b⊥ ∩ a)) = ((a ∩ b) ∪ (a ∩ b⊥ ))
8	7	ax-r5 38	. . . . 5 (((b ∩ a) ∪ (b⊥ ∩ a)) ∪ a⊥ ) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ )
9		ax-a2 31	. . . . 5 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
10	8, 9	ax-r2 36	. . . 4 (((b ∩ a) ∪ (b⊥ ∩ a)) ∪ a⊥ ) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
11	4, 10	ax-r2 36	. . 3 ((b →4 a) ∪ a⊥ ) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
12	3, 11	ax-r2 36	. 2 (a⊥ ∪ (b →4 a)) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
13	2, 12	ax-r2 36	1 (a →4 (b →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:  u4lem3n  755  u4lem4  759