Theorem ud3 597

Description: Unified disjunction for Kalmbach implication.

Assertion

Ref	Expression
ud3	(a ∪ b) = ((a →3 b) →3 (((a →3 b) →3 (b →3 a)) →3 a))

Proof of Theorem ud3

Step	Hyp	Ref	Expression
1		ud3lem1 570	. . . . . 6 ((a →3 b) →3 (b →3 a)) = (a ∪ (a⊥ ∩ b⊥ ))
2	1	ud3lem0b 261	. . . . 5 (((a →3 b) →3 (b →3 a)) →3 a) = ((a ∪ (a⊥ ∩ b⊥ )) →3 a)
3		ud3lem2 571	. . . . 5 ((a ∪ (a⊥ ∩ b⊥ )) →3 a) = (a ∪ b)
4	2, 3	ax-r2 36	. . . 4 (((a →3 b) →3 (b →3 a)) →3 a) = (a ∪ b)
5	4	ud3lem0a 260	. . 3 ((a →3 b) →3 (((a →3 b) →3 (b →3 a)) →3 a)) = ((a →3 b) →3 (a ∪ b))
6		ud3lem3 576	. . 3 ((a →3 b) →3 (a ∪ b)) = (a ∪ b)
7	5, 6	ax-r2 36	. 2 ((a →3 b) →3 (((a →3 b) →3 (b →3 a)) →3 a)) = (a ∪ b)
8	7	ax-r1 35	1 (a ∪ b) = ((a →3 b) →3 (((a →3 b) →3 (b →3 a)) →3 a))

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

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