Theorem i3n1 241

Description: Equivalence for Kalmbach implication.

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 45	. 2 (a⊥ →3 b⊥ ) = (((a⊥ ⊥ ∩ b⊥ ) ∪ (a⊥ ⊥ ∩ b⊥ ⊥ )) ∪ (a⊥ ∩ (a⊥ ⊥ ∪ b⊥ )))
2		ax-a1 29	. . . . . 6 a = a⊥ ⊥
3	2	ran 71	. . . . 5 (a ∩ b⊥ ) = (a⊥ ⊥ ∩ b⊥ )
4		ax-a1 29	. . . . . 6 b = b⊥ ⊥
5	2, 4	2an 72	. . . . 5 (a ∩ b) = (a⊥ ⊥ ∩ b⊥ ⊥ )
6	3, 5	2or 67	. . . 4 ((a ∩ b⊥ ) ∪ (a ∩ b)) = ((a⊥ ⊥ ∩ b⊥ ) ∪ (a⊥ ⊥ ∩ b⊥ ⊥ ))
7	2	ax-r5 37	. . . . 5 (a ∪ b⊥ ) = (a⊥ ⊥ ∪ b⊥ )
8	7	lan 70	. . . 4 (a⊥ ∩ (a ∪ b⊥ )) = (a⊥ ∩ (a⊥ ⊥ ∪ b⊥ ))
9	6, 8	2or 67	. . 3 (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ ))) = (((a⊥ ⊥ ∩ b⊥ ) ∪ (a⊥ ⊥ ∩ b⊥ ⊥ )) ∪ (a⊥ ∩ (a⊥ ⊥ ∪ b⊥ )))
10	9	ax-r1 34	. 2 (((a⊥ ⊥ ∩ b⊥ ) ∪ (a⊥ ⊥ ∩ b⊥ ⊥ )) ∪ (a⊥ ∩ (a⊥ ⊥ ∪ b⊥ ))) = (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ )))
11	1, 10	ax-r2 35	1 (a⊥ →3 b⊥ ) = (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 15

This theorem is referenced by:  oi3ai3 485  i3con 533  i3orlem7 540  i3orlem8 541

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-i3 45

metamath.org