Theorem u3lemnona 667

Description: Lemma for Kalmbach implication study. (Contributed by NM, 16-Dec-1997.)

Assertion

Ref	Expression
u3lemnona	((a →3 b)⊥ ∪ a⊥ ) = (a⊥ ∪ (a ∩ b⊥ ))

Proof of Theorem u3lemnona

Step	Hyp	Ref	Expression
1		u3lemaa 602	. . . 4 ((a →3 b) ∩ a) = (a ∩ (a⊥ ∪ b))
2		oran2 92	. . . . 5 (a⊥ ∪ b) = (a ∩ b⊥ )⊥
3	2	lan 77	. . . 4 (a ∩ (a⊥ ∪ b)) = (a ∩ (a ∩ b⊥ )⊥ )
4	1, 3	ax-r2 36	. . 3 ((a →3 b) ∩ a) = (a ∩ (a ∩ b⊥ )⊥ )
5		df-a 40	. . 3 ((a →3 b) ∩ a) = ((a →3 b)⊥ ∪ a⊥ )⊥
6		anor1 88	. . 3 (a ∩ (a ∩ b⊥ )⊥ ) = (a⊥ ∪ (a ∩ b⊥ ))⊥
7	4, 5, 6	3tr2 64	. 2 ((a →3 b)⊥ ∪ a⊥ )⊥ = (a⊥ ∪ (a ∩ b⊥ ))⊥
8	7	con1 66	1 ((a →3 b)⊥ ∪ a⊥ ) = (a⊥ ∪ (a ∩ b⊥ ))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u3lem13b  790