Theorem ud5 599

Description: Unified disjunction for relevance implication.

Assertion

Ref	Expression
ud5	(a ∪ b) = ((a →5 b) →5 (((a →5 b) →5 (b →5 a)) →5 a))

Proof of Theorem ud5

Step	Hyp	Ref	Expression
1		ud5lem1 589	. . . . . 6 ((a →5 b) →5 (b →5 a)) = (a ∪ b⊥ )
2	1	ud5lem0b 265	. . . . 5 (((a →5 b) →5 (b →5 a)) →5 a) = ((a ∪ b⊥ ) →5 a)
3		ud5lem2 590	. . . . 5 ((a ∪ b⊥ ) →5 a) = (a ∪ (a⊥ ∩ b))
4	2, 3	ax-r2 36	. . . 4 (((a →5 b) →5 (b →5 a)) →5 a) = (a ∪ (a⊥ ∩ b))
5	4	ud5lem0a 264	. . 3 ((a →5 b) →5 (((a →5 b) →5 (b →5 a)) →5 a)) = ((a →5 b) →5 (a ∪ (a⊥ ∩ b)))
6		ud5lem3 594	. . 3 ((a →5 b) →5 (a ∪ (a⊥ ∩ b))) = (a ∪ b)
7	5, 6	ax-r2 36	. 2 ((a →5 b) →5 (((a →5 b) →5 (b →5 a)) →5 a)) = (a ∪ b)
8	7	ax-r1 35	1 (a ∪ b) = ((a →5 b) →5 (((a →5 b) →5 (b →5 a)) →5 a))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 16

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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133

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