Theorem i3n2 501

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

Assertion

Ref	Expression
i3n2	(a⊥ →3 b⊥ ) = ((a ∩ b) ∪ ((a ∪ b⊥ ) ∩ (a⊥ ∪ (a ∩ b⊥ ))))

Proof of Theorem i3n2

Step	Hyp	Ref	Expression
1		df2i3 498	. 2 (a⊥ →3 b⊥ ) = ((a⊥ ⊥ ∩ b⊥ ⊥ ) ∪ ((a⊥ ⊥ ∪ b⊥ ) ∩ (a⊥ ∪ (a⊥ ⊥ ∩ b⊥ ))))
2		ax-a1 30	. . . . 5 a = a⊥ ⊥
3		ax-a1 30	. . . . 5 b = b⊥ ⊥
4	2, 3	2an 79	. . . 4 (a ∩ b) = (a⊥ ⊥ ∩ b⊥ ⊥ )
5	2	ax-r5 38	. . . . 5 (a ∪ b⊥ ) = (a⊥ ⊥ ∪ b⊥ )
6	2	ran 78	. . . . . 6 (a ∩ b⊥ ) = (a⊥ ⊥ ∩ b⊥ )
7	6	lor 70	. . . . 5 (a⊥ ∪ (a ∩ b⊥ )) = (a⊥ ∪ (a⊥ ⊥ ∩ b⊥ ))
8	5, 7	2an 79	. . . 4 ((a ∪ b⊥ ) ∩ (a⊥ ∪ (a ∩ b⊥ ))) = ((a⊥ ⊥ ∪ b⊥ ) ∩ (a⊥ ∪ (a⊥ ⊥ ∩ b⊥ )))
9	4, 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⊥ ))))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)