Theorem ud3lem3a 572

Description: Lemma for unified disjunction. (Contributed by NM, 27-Nov-1997.)

Assertion

Ref	Expression
ud3lem3a	((a →3 b)⊥ ∩ (a ∪ b)) = (a →3 b)⊥

Proof of Theorem ud3lem3a

Step	Hyp	Ref	Expression
1		ud3lem0c 279	. . 3 (a →3 b)⊥ = (((a ∪ b⊥ ) ∩ (a ∪ b)) ∩ (a⊥ ∪ (a ∩ b⊥ )))
2		lea 160	. . . 4 (((a ∪ b⊥ ) ∩ (a ∪ b)) ∩ (a⊥ ∪ (a ∩ b⊥ ))) ≤ ((a ∪ b⊥ ) ∩ (a ∪ b))
3		lear 161	. . . 4 ((a ∪ b⊥ ) ∩ (a ∪ b)) ≤ (a ∪ b)
4	2, 3	letr 137	. . 3 (((a ∪ b⊥ ) ∩ (a ∪ b)) ∩ (a⊥ ∪ (a ∩ b⊥ ))) ≤ (a ∪ b)
5	1, 4	bltr 138	. 2 (a →3 b)⊥ ≤ (a ∪ b)
6	5	df2le2 136	1 ((a →3 b)⊥ ∩ (a ∪ b)) = (a →3 b)⊥

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

This proof depends on 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 proof depends on definitions:  df-a 40  df-i3 46  df-le1 130  df-le2 131

This theorem is used by:  ud3lem3  576