Theorem u3lem2 728

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u3lem2	(((a →3 b) →3 a) →3 a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Proof of Theorem u3lem2

Step	Hyp	Ref	Expression
1		comi31 490	. . . . 5 a C (a →3 b)
2		comid 179	. . . . 5 a C a
3	1, 2	u3lemc2 670	. . . 4 a C ((a →3 b) →3 a)
4	3	comcom 435	. . 3 ((a →3 b) →3 a) C a
5	4	u3lemc4 685	. 2 (((a →3 b) →3 a) →3 a) = (((a →3 b) →3 a)⊥ ∪ a)
6		u3lem1n 723	. . . 4 ((a →3 b) →3 a)⊥ = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
7	6	ax-r5 37	. . 3 (((a →3 b) →3 a)⊥ ∪ a) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
8		ax-a2 30	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
9	7, 8	ax-r2 35	. 2 (((a →3 b) →3 a)⊥ ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
10	5, 9	ax-r2 35	1 (((a →3 b) →3 a) →3 a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

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

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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