Theorem u3lemnona 649

Description: Lemma for Kalmbach implication study.

Assertion

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

Proof of Theorem u3lemnona

Step	Hyp	Ref	Expression
1		u3lemaa 584	. . . 4 ((a →3 b) ∩ a) = (a ∩ (a⊥ ∪ b))
2		oran2 84	. . . . 5 (a⊥ ∪ b) = (a ∩ b⊥ )⊥
3	2	lan 70	. . . 4 (a ∩ (a⊥ ∪ b)) = (a ∩ (a ∩ b⊥ )⊥ )
4	1, 3	ax-r2 35	. . 3 ((a →3 b) ∩ a) = (a ∩ (a ∩ b⊥ )⊥ )
5		df-a 39	. . 3 ((a →3 b) ∩ a) = ((a →3 b)⊥ ∪ a⊥ )⊥
6		anor1 80	. . 3 (a ∩ (a ∩ b⊥ )⊥ ) = (a⊥ ∪ (a ∩ b⊥ ))⊥
7	4, 5, 6	3tr2 61	. 2 ((a →3 b)⊥ ∪ a⊥ )⊥ = (a⊥ ∪ (a ∩ b⊥ ))⊥
8	7	con1 63	1 ((a →3 b)⊥ ∪ a⊥ ) = (a⊥ ∪ (a ∩ b⊥ ))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 15

This theorem is referenced by:  u3lem13b 772

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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