Theorem u5lemona 611

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemona	((a →5 b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))

Proof of Theorem u5lemona

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) ∪ a⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ )
3		ax-a3 31	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
4	3	ax-r5 37	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) = (((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a⊥ )
5		ax-a3 31	. . . 4 (((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a⊥ ) = ((a ∩ b) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ))
6		lea 152	. . . . . . . 8 (a⊥ ∩ b) ≤ a⊥
7		lea 152	. . . . . . . 8 (a⊥ ∩ b⊥ ) ≤ a⊥
8	6, 7	lel2or 162	. . . . . . 7 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
9	8	df-le2 123	. . . . . 6 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) = a⊥
10	9	lor 66	. . . . 5 ((a ∩ b) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ )) = ((a ∩ b) ∪ a⊥ )
11		ax-a2 30	. . . . 5 ((a ∩ b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
12	10, 11	ax-r2 35	. . . 4 ((a ∩ b) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ )) = (a⊥ ∪ (a ∩ b))
13	5, 12	ax-r2 35	. . 3 (((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
14	4, 13	ax-r2 35	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
15	2, 14	ax-r2 35	1 ((a →5 b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))

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

This theorem is referenced by:  u5lemnaa 626  u5lem5 747

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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