Theorem u5lemonb 639

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

Assertion

Ref	Expression
u5lemonb	((a →5 b) ∪ b⊥ ) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b⊥ )

Proof of Theorem u5lemonb

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) ∪ b⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ )
3		ax-a3 32	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∩ b⊥ ) ∪ b⊥ ))
4		lear 161	. . . . 5 (a⊥ ∩ b⊥ ) ≤ b⊥
5	4	df-le2 131	. . . 4 ((a⊥ ∩ b⊥ ) ∪ b⊥ ) = b⊥
6	5	lor 70	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∩ b⊥ ) ∪ b⊥ )) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b⊥ )
7	3, 6	ax-r2 36	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b⊥ )
8	2, 7	ax-r2 36	1 ((a →5 b) ∪ b⊥ ) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b⊥ )

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 16

This proof depends on axioms:  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-i5 48  df-le1 130  df-le2 131

This theorem is used by:  u5lemnab  654  u5lem3  753