Theorem u3lem2 746

Description: Lemma for unified implication study. (Contributed by NM, 24-Dec-1997.)

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 508	. . . . 5 a C (a →3 b)
2		comid 187	. . . . 5 a C a
3	1, 2	u3lemc2 688	. . . 4 a C ((a →3 b) →3 a)
4	3	comcom 453	. . 3 ((a →3 b) →3 a) C a
5	4	u3lemc4 703	. 2 (((a →3 b) →3 a) →3 a) = (((a →3 b) →3 a)⊥ ∪ a)
6		u3lem1n 741	. . . 4 ((a →3 b) →3 a)⊥ = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
7	6	ax-r5 38	. . 3 (((a →3 b) →3 a)⊥ ∪ a) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
8		ax-a2 31	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
9	7, 8	ax-r2 36	. 2 (((a →3 b) →3 a)⊥ ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
10	5, 9	ax-r2 36	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 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: (None)