Theorem u5lemnaa 626

Description: Lemma for relevance implication study.

Assertion

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

Proof of Theorem u5lemnaa

Step	Hyp	Ref	Expression
1		anor2 81	. 2 ((a →5 b)⊥ ∩ a) = ((a →5 b) ∪ a⊥ )⊥
2		u5lemona 611	. . . 4 ((a →5 b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
3	2	ax-r4 36	. . 3 ((a →5 b) ∪ a⊥ )⊥ = (a⊥ ∪ (a ∩ b))⊥
4		anor1 80	. . . . 5 (a ∩ (a ∩ b)⊥ ) = (a⊥ ∪ (a ∩ b))⊥
5	4	ax-r1 34	. . . 4 (a⊥ ∪ (a ∩ b))⊥ = (a ∩ (a ∩ b)⊥ )
6		df-a 39	. . . . . 6 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
7	6	con2 64	. . . . 5 (a ∩ b)⊥ = (a⊥ ∪ b⊥ )
8	7	lan 70	. . . 4 (a ∩ (a ∩ b)⊥ ) = (a ∩ (a⊥ ∪ b⊥ ))
9	5, 8	ax-r2 35	. . 3 (a⊥ ∪ (a ∩ b))⊥ = (a ∩ (a⊥ ∪ b⊥ ))
10	3, 9	ax-r2 35	. 2 ((a →5 b) ∪ a⊥ )⊥ = (a ∩ (a⊥ ∪ b⊥ ))
11	1, 10	ax-r2 35	1 ((a →5 b)⊥ ∩ a) = (a ∩ (a⊥ ∪ b⊥ ))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 17

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i5 47  df-le1 122  df-le2 123

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