Theorem ud4 580

Description: Unified disjunction for non-tollens implication.

Assertion

Ref	Expression
ud4	(a ∪ b) = ((a →4 b) →4 (((a →4 b) →4 (b →4 a)) →4 a))

Proof of Theorem ud4

Step	Hyp	Ref	Expression
1		ud4lem1 563	. . . . . 6 ((a →4 b) →4 (b →4 a)) = (a ∪ (a⊥ ∩ b⊥ ))
2	1	ud4lem0b 255	. . . . 5 (((a →4 b) →4 (b →4 a)) →4 a) = ((a ∪ (a⊥ ∩ b⊥ )) →4 a)
3		ud4lem2 564	. . . . 5 ((a ∪ (a⊥ ∩ b⊥ )) →4 a) = (a ∪ b)
4	2, 3	ax-r2 35	. . . 4 (((a →4 b) →4 (b →4 a)) →4 a) = (a ∪ b)
5	4	ud4lem0a 254	. . 3 ((a →4 b) →4 (((a →4 b) →4 (b →4 a)) →4 a)) = ((a →4 b) →4 (a ∪ b))
6		ud4lem3 567	. . 3 ((a →4 b) →4 (a ∪ b)) = (a ∪ b)
7	5, 6	ax-r2 35	. 2 ((a →4 b) →4 (((a →4 b) →4 (b →4 a)) →4 a)) = (a ∪ b)
8	7	ax-r1 34	1 (a ∪ b) = ((a →4 b) →4 (((a →4 b) →4 (b →4 a)) →4 a))

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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