Theorem u2lemob 631

Description: Lemma for Dishkant implication study. (Contributed by NM, 15-Dec-1997.)

Assertion

Ref	Expression
u2lemob	((a →2 b) ∪ b) = ((a⊥ ∩ b⊥ ) ∪ b)

Proof of Theorem u2lemob

Step	Hyp	Ref	Expression
1		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2	1	ax-r5 38	. 2 ((a →2 b) ∪ b) = ((b ∪ (a⊥ ∩ b⊥ )) ∪ b)
3		or32 82	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∪ b) = ((b ∪ b) ∪ (a⊥ ∩ b⊥ ))
4		ax-a2 31	. . . 4 ((b ∪ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (b ∪ b))
5		oridm 110	. . . . 5 (b ∪ b) = b
6	5	lor 70	. . . 4 ((a⊥ ∩ b⊥ ) ∪ (b ∪ b)) = ((a⊥ ∩ b⊥ ) ∪ b)
7	4, 6	ax-r2 36	. . 3 ((b ∪ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ b)
8	3, 7	ax-r2 36	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∪ b) = ((a⊥ ∩ b⊥ ) ∪ b)
9	2, 8	ax-r2 36	1 ((a →2 b) ∪ b) = ((a⊥ ∩ b⊥ ) ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-t 41  df-f 42  df-i2 45

This theorem is referenced by:  u2lemnanb  656