Theorem ud1lem0c 277

Description: Lemma for unified disjunction. (Contributed by NM, 23-Nov-1997.)

Assertion

Ref	Expression
ud1lem0c	(a →1 b)⊥ = (a ∩ (a⊥ ∪ b⊥ ))

Proof of Theorem ud1lem0c

Step	Hyp	Ref	Expression
1		df-i1 44	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
2		df-a 40	. . . . . 6 (a ∩ (a⊥ ∪ b⊥ )) = (a⊥ ∪ (a⊥ ∪ b⊥ )⊥ )⊥
3		df-a 40	. . . . . . . . 9 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
4	3	ax-r1 35	. . . . . . . 8 (a⊥ ∪ b⊥ )⊥ = (a ∩ b)
5	4	lor 70	. . . . . . 7 (a⊥ ∪ (a⊥ ∪ b⊥ )⊥ ) = (a⊥ ∪ (a ∩ b))
6	5	ax-r4 37	. . . . . 6 (a⊥ ∪ (a⊥ ∪ b⊥ )⊥ )⊥ = (a⊥ ∪ (a ∩ b))⊥
7	2, 6	ax-r2 36	. . . . 5 (a ∩ (a⊥ ∪ b⊥ )) = (a⊥ ∪ (a ∩ b))⊥
8	7	ax-r1 35	. . . 4 (a⊥ ∪ (a ∩ b))⊥ = (a ∩ (a⊥ ∪ b⊥ ))
9	8	con3 68	. . 3 (a⊥ ∪ (a ∩ b)) = (a ∩ (a⊥ ∪ b⊥ ))⊥
10	1, 9	ax-r2 36	. 2 (a →1 b) = (a ∩ (a⊥ ∪ b⊥ ))⊥
11	10	con2 67	1 (a →1 b)⊥ = (a ∩ (a⊥ ∪ b⊥ ))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

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

This theorem depends on definitions:  df-a 40  df-i1 44

This theorem is referenced by:  ud1lem1  560  ud1lem3  562  u1lemc6  706  u1lem11  780  i1abs  801  sa5  836  elimcons2  869  kb10iii  893