Theorem dfi4b 500

Description: Alternate non-tollens conditional. (Contributed by NM, 6-Aug-2001.)

Assertion

Ref	Expression
dfi4b	(a →4 b) = ((a⊥ ∪ b) ∩ ((b⊥ ∪ (b ∩ a⊥ )) ∪ (b ∩ a)))

Proof of Theorem dfi4b

Step	Hyp	Ref	Expression
1		i4i3 271	. 2 (a →4 b) = (b⊥ →3 a⊥ )
2		dfi3b 499	. . 3 (b⊥ →3 a⊥ ) = ((b⊥ ⊥ ∪ a⊥ ) ∩ ((b⊥ ∪ (b⊥ ⊥ ∩ a⊥ ⊥ )) ∪ (b⊥ ⊥ ∩ a⊥ )))
3		ax-a2 31	. . . . . 6 (a⊥ ∪ b) = (b ∪ a⊥ )
4		ax-a1 30	. . . . . . 7 b = b⊥ ⊥
5	4	ax-r5 38	. . . . . 6 (b ∪ a⊥ ) = (b⊥ ⊥ ∪ a⊥ )
6	3, 5	ax-r2 36	. . . . 5 (a⊥ ∪ b) = (b⊥ ⊥ ∪ a⊥ )
7	4	ran 78	. . . . . . . 8 (b ∩ a⊥ ) = (b⊥ ⊥ ∩ a⊥ )
8	7	lor 70	. . . . . . 7 (b⊥ ∪ (b ∩ a⊥ )) = (b⊥ ∪ (b⊥ ⊥ ∩ a⊥ ))
9		ax-a1 30	. . . . . . . 8 a = a⊥ ⊥
10	4, 9	2an 79	. . . . . . 7 (b ∩ a) = (b⊥ ⊥ ∩ a⊥ ⊥ )
11	8, 10	2or 72	. . . . . 6 ((b⊥ ∪ (b ∩ a⊥ )) ∪ (b ∩ a)) = ((b⊥ ∪ (b⊥ ⊥ ∩ a⊥ )) ∪ (b⊥ ⊥ ∩ a⊥ ⊥ ))
12		or32 82	. . . . . 6 ((b⊥ ∪ (b⊥ ⊥ ∩ a⊥ )) ∪ (b⊥ ⊥ ∩ a⊥ ⊥ )) = ((b⊥ ∪ (b⊥ ⊥ ∩ a⊥ ⊥ )) ∪ (b⊥ ⊥ ∩ a⊥ ))
13	11, 12	ax-r2 36	. . . . 5 ((b⊥ ∪ (b ∩ a⊥ )) ∪ (b ∩ a)) = ((b⊥ ∪ (b⊥ ⊥ ∩ a⊥ ⊥ )) ∪ (b⊥ ⊥ ∩ a⊥ ))
14	6, 13	2an 79	. . . 4 ((a⊥ ∪ b) ∩ ((b⊥ ∪ (b ∩ a⊥ )) ∪ (b ∩ a))) = ((b⊥ ⊥ ∪ a⊥ ) ∩ ((b⊥ ∪ (b⊥ ⊥ ∩ a⊥ ⊥ )) ∪ (b⊥ ⊥ ∩ a⊥ )))
15	14	ax-r1 35	. . 3 ((b⊥ ⊥ ∪ a⊥ ) ∩ ((b⊥ ∪ (b⊥ ⊥ ∩ a⊥ ⊥ )) ∪ (b⊥ ⊥ ∩ a⊥ ))) = ((a⊥ ∪ b) ∩ ((b⊥ ∪ (b ∩ a⊥ )) ∪ (b ∩ a)))
16	2, 15	ax-r2 36	. 2 (b⊥ →3 a⊥ ) = ((a⊥ ∪ b) ∩ ((b⊥ ∪ (b ∩ a⊥ )) ∪ (b ∩ a)))
17	1, 16	ax-r2 36	1 (a →4 b) = ((a⊥ ∪ b) ∩ ((b⊥ ∪ (b ∩ a⊥ )) ∪ (b ∩ a)))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14   →₄ wi4 15

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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  negantlem10  861