Theorem u5lemona 629

Description: Lemma for relevance implication study. (Contributed by NM, 15-Dec-1997.)

Assertion

Ref	Expression
u5lemona	((a →5 b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))

Proof of Theorem u5lemona

Step	Hyp	Ref	Expression
1		df-i5 48	. . 3 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
2	1	ax-r5 38	. 2 ((a →5 b) ∪ a⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ )
3		ax-a3 32	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
4	3	ax-r5 38	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) = (((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a⊥ )
5		ax-a3 32	. . . 4 (((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a⊥ ) = ((a ∩ b) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ))
6		lea 160	. . . . . . . 8 (a⊥ ∩ b) ≤ a⊥
7		lea 160	. . . . . . . 8 (a⊥ ∩ b⊥ ) ≤ a⊥
8	6, 7	lel2or 170	. . . . . . 7 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
9	8	df-le2 131	. . . . . 6 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) = a⊥
10	9	lor 70	. . . . 5 ((a ∩ b) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ )) = ((a ∩ b) ∪ a⊥ )
11		ax-a2 31	. . . . 5 ((a ∩ b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
12	10, 11	ax-r2 36	. . . 4 ((a ∩ b) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ )) = (a⊥ ∪ (a ∩ b))
13	5, 12	ax-r2 36	. . 3 (((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
14	4, 13	ax-r2 36	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
15	2, 14	ax-r2 36	1 ((a →5 b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 16

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

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i5 48  df-le1 130  df-le2 131

This theorem is referenced by:  u5lemnaa  644  u5lem5  765