Theorem i3n1 249

Description: Equivalence for Kalmbach implication. (Contributed by NM, 9-Nov-1997.)

Assertion

Ref	Expression
i3n1	(a⊥ →3 b⊥ ) = (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ )))

Proof of Theorem i3n1

Step	Hyp	Ref	Expression
1		df-i3 46	. 2 (a⊥ →3 b⊥ ) = (((a⊥ ⊥ ∩ b⊥ ) ∪ (a⊥ ⊥ ∩ b⊥ ⊥ )) ∪ (a⊥ ∩ (a⊥ ⊥ ∪ b⊥ )))
2		ax-a1 30	. . . . . 6 a = a⊥ ⊥
3	2	ran 78	. . . . 5 (a ∩ b⊥ ) = (a⊥ ⊥ ∩ b⊥ )
4		ax-a1 30	. . . . . 6 b = b⊥ ⊥
5	2, 4	2an 79	. . . . 5 (a ∩ b) = (a⊥ ⊥ ∩ b⊥ ⊥ )
6	3, 5	2or 72	. . . 4 ((a ∩ b⊥ ) ∪ (a ∩ b)) = ((a⊥ ⊥ ∩ b⊥ ) ∪ (a⊥ ⊥ ∩ b⊥ ⊥ ))
7	2	ax-r5 38	. . . . 5 (a ∪ b⊥ ) = (a⊥ ⊥ ∪ b⊥ )
8	7	lan 77	. . . 4 (a⊥ ∩ (a ∪ b⊥ )) = (a⊥ ∩ (a⊥ ⊥ ∪ b⊥ ))
9	6, 8	2or 72	. . 3 (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ ))) = (((a⊥ ⊥ ∩ b⊥ ) ∪ (a⊥ ⊥ ∩ b⊥ ⊥ )) ∪ (a⊥ ∩ (a⊥ ⊥ ∪ b⊥ )))
10	9	ax-r1 35	. 2 (((a⊥ ⊥ ∩ b⊥ ) ∪ (a⊥ ⊥ ∩ b⊥ ⊥ )) ∪ (a⊥ ∩ (a⊥ ⊥ ∪ b⊥ ))) = (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ )))
11	1, 10	ax-r2 36	1 (a⊥ →3 b⊥ ) = (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ 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-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-i3 46

This theorem is used by:  oi3ai3  503  i3con  551  i3orlem7  558  i3orlem8  559