Theorem ud1lem0c 269

Description: Lemma for unified disjunction.

Assertion

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

Proof of Theorem ud1lem0c

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
2		df-a 39	. . . . . 6 (a ∩ (a⊥ ∪ b⊥ )) = (a⊥ ∪ (a⊥ ∪ b⊥ )⊥ )⊥
3		df-a 39	. . . . . . . . 9 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
4	3	ax-r1 34	. . . . . . . 8 (a⊥ ∪ b⊥ )⊥ = (a ∩ b)
5	4	lor 66	. . . . . . 7 (a⊥ ∪ (a⊥ ∪ b⊥ )⊥ ) = (a⊥ ∪ (a ∩ b))
6	5	ax-r4 36	. . . . . 6 (a⊥ ∪ (a⊥ ∪ b⊥ )⊥ )⊥ = (a⊥ ∪ (a ∩ b))⊥
7	2, 6	ax-r2 35	. . . . 5 (a ∩ (a⊥ ∪ b⊥ )) = (a⊥ ∪ (a ∩ b))⊥
8	7	ax-r1 34	. . . 4 (a⊥ ∪ (a ∩ b))⊥ = (a ∩ (a⊥ ∪ b⊥ ))
9	8	con3 65	. . 3 (a⊥ ∪ (a ∩ b)) = (a ∩ (a⊥ ∪ b⊥ ))⊥
10	1, 9	ax-r2 35	. 2 (a →1 b) = (a ∩ (a⊥ ∪ b⊥ ))⊥
11	10	con2 64	1 (a →1 b)⊥ = (a ∩ (a⊥ ∪ b⊥ ))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  ud1lem1 542  ud1lem3 544  u1lemc6 688  u1lem11 762  i1abs 783  sa5 818  elimcons2 851  kb10iii 875

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

This theorem depends on definitions:  df-a 39  df-i1 43

metamath.org