Theorem u2lemonb 618

Description: Lemma for Dishkant implication study.

Assertion

Ref	Expression
u2lemonb	((a →2 b) ∪ b⊥ ) = 1

Proof of Theorem u2lemonb

Step	Hyp	Ref	Expression
1		df-i2 44	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2	1	ax-r5 37	. 2 ((a →2 b) ∪ b⊥ ) = ((b ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ )
3		or32 75	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) = ((b ∪ b⊥ ) ∪ (a⊥ ∩ b⊥ ))
4		ax-a2 30	. . . 4 ((b ∪ b⊥ ) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (b ∪ b⊥ ))
5		df-t 40	. . . . . . 7 1 = (b ∪ b⊥ )
6	5	lor 66	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ 1) = ((a⊥ ∩ b⊥ ) ∪ (b ∪ b⊥ ))
7	6	ax-r1 34	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ (b ∪ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ 1)
8		or1 96	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ 1) = 1
9	7, 8	ax-r2 35	. . . 4 ((a⊥ ∩ b⊥ ) ∪ (b ∪ b⊥ )) = 1
10	4, 9	ax-r2 35	. . 3 ((b ∪ b⊥ ) ∪ (a⊥ ∩ b⊥ )) = 1
11	3, 10	ax-r2 35	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) = 1
12	2, 11	ax-r2 35	1 ((a →2 b) ∪ b⊥ ) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₂ wi2 14

This theorem is referenced by:  u2lemnab 633  u2lem3 732  oa23 916

This theorem was proved from axioms:  ax-a2 30  ax-a3 31  ax-a4 32  ax-r1 34  ax-r2 35  ax-r5 37

This theorem depends on definitions:  df-t 40  df-i2 44

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