Theorem u5lemnaa 644

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

Assertion

Ref	Expression
u5lemnaa	((a →5 b)⊥ ∩ a) = (a ∩ (a⊥ ∪ b⊥ ))

Proof of Theorem u5lemnaa

Step	Hyp	Ref	Expression
1		anor2 89	. 2 ((a →5 b)⊥ ∩ a) = ((a →5 b) ∪ a⊥ )⊥
2		u5lemona 629	. . . 4 ((a →5 b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
3	2	ax-r4 37	. . 3 ((a →5 b) ∪ a⊥ )⊥ = (a⊥ ∪ (a ∩ b))⊥
4		anor1 88	. . . . 5 (a ∩ (a ∩ b)⊥ ) = (a⊥ ∪ (a ∩ b))⊥
5	4	ax-r1 35	. . . 4 (a⊥ ∪ (a ∩ b))⊥ = (a ∩ (a ∩ b)⊥ )
6		df-a 40	. . . . . 6 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
7	6	con2 67	. . . . 5 (a ∩ b)⊥ = (a⊥ ∪ b⊥ )
8	7	lan 77	. . . 4 (a ∩ (a ∩ b)⊥ ) = (a ∩ (a⊥ ∪ b⊥ ))
9	5, 8	ax-r2 36	. . 3 (a⊥ ∪ (a ∩ b))⊥ = (a ∩ (a⊥ ∪ b⊥ ))
10	3, 9	ax-r2 36	. 2 ((a →5 b) ∪ a⊥ )⊥ = (a ∩ (a⊥ ∪ b⊥ ))
11	1, 10	ax-r2 36	1 ((a →5 b)⊥ ∩ a) = (a ∩ (a⊥ ∪ b⊥ ))

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

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

This theorem is used by: (None)