Theorem u5lemob 616

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemob	((a →5 b) ∪ b) = ((a⊥ ∩ b⊥ ) ∪ b)

Proof of Theorem u5lemob

Step	Hyp	Ref	Expression
1		df-i5 47	. . 3 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
2	1	ax-r5 37	. 2 ((a →5 b) ∪ b) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ b)
3		ax-a3 31	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∩ b⊥ ) ∪ b))
4		lear 153	. . . . . 6 (a ∩ b) ≤ b
5		lear 153	. . . . . 6 (a⊥ ∩ b) ≤ b
6	4, 5	lel2or 162	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b)) ≤ b
7		leor 151	. . . . 5 b ≤ ((a⊥ ∩ b⊥ ) ∪ b)
8	6, 7	letr 129	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b)) ≤ ((a⊥ ∩ b⊥ ) ∪ b)
9	8	df-le2 123	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∩ b⊥ ) ∪ b)) = ((a⊥ ∩ b⊥ ) ∪ b)
10	3, 9	ax-r2 35	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ b) = ((a⊥ ∩ b⊥ ) ∪ b)
11	2, 10	ax-r2 35	1 ((a →5 b) ∪ b) = ((a⊥ ∩ b⊥ ) ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 17

This theorem is referenced by:  u5lemnanb 641

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i5 47  df-le1 122  df-le2 123

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