Theorem u4lem3n 737

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u4lem3n	(a →4 (b →4 a))⊥ = (a ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))

Proof of Theorem u4lem3n

Step	Hyp	Ref	Expression
1		u4lem3 734	. . 3 (a →4 (b →4 a)) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
2		ax-a2 30	. . . . . 6 ((a ∩ b) ∪ (a ∩ b⊥ )) = ((a ∩ b⊥ ) ∪ (a ∩ b))
3		anor1 80	. . . . . . . 8 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
4		df-a 39	. . . . . . . 8 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
5	3, 4	2or 67	. . . . . . 7 ((a ∩ b⊥ ) ∪ (a ∩ b)) = ((a⊥ ∪ b)⊥ ∪ (a⊥ ∪ b⊥ )⊥ )
6		oran3 85	. . . . . . 7 ((a⊥ ∪ b)⊥ ∪ (a⊥ ∪ b⊥ )⊥ ) = ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ ))⊥
7	5, 6	ax-r2 35	. . . . . 6 ((a ∩ b⊥ ) ∪ (a ∩ b)) = ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ ))⊥
8	2, 7	ax-r2 35	. . . . 5 ((a ∩ b) ∪ (a ∩ b⊥ )) = ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ ))⊥
9	8	lor 66	. . . 4 (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a⊥ ∪ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ ))⊥ )
10		oran3 85	. . . 4 (a⊥ ∪ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ ))⊥ ) = (a ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))⊥
11	9, 10	ax-r2 35	. . 3 (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))⊥
12	1, 11	ax-r2 35	. 2 (a →4 (b →4 a)) = (a ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))⊥
13	12	con2 64	1 (a →4 (b →4 a))⊥ = (a ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 16

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-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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