Theorem u4lemc2 689

Description: Commutation theorem for non-tollens implication. (Contributed by NM, 14-Dec-1997.)

Hypotheses

Ref	Expression
ulemc2.1	a C b
ulemc2.2	a C c

Assertion

Ref	Expression
u4lemc2	a C (b →4 c)

Proof of Theorem u4lemc2

Step	Hyp	Ref	Expression
1		ulemc2.1	. . . . 5 a C b
2		ulemc2.2	. . . . 5 a C c
3	1, 2	com2an 484	. . . 4 a C (b ∩ c)
4	1	comcom2 183	. . . . 5 a C b⊥
5	4, 2	com2an 484	. . . 4 a C (b⊥ ∩ c)
6	3, 5	com2or 483	. . 3 a C ((b ∩ c) ∪ (b⊥ ∩ c))
7	4, 2	com2or 483	. . . 4 a C (b⊥ ∪ c)
8	2	comcom2 183	. . . 4 a C c⊥
9	7, 8	com2an 484	. . 3 a C ((b⊥ ∪ c) ∩ c⊥ )
10	6, 9	com2or 483	. 2 a C (((b ∩ c) ∪ (b⊥ ∩ c)) ∪ ((b⊥ ∪ c) ∩ c⊥ ))
11		df-i4 47	. . 3 (b →4 c) = (((b ∩ c) ∪ (b⊥ ∩ c)) ∪ ((b⊥ ∪ c) ∩ c⊥ ))
12	11	ax-r1 35	. 2 (((b ∩ c) ∪ (b⊥ ∩ c)) ∪ ((b⊥ ∪ c) ∩ c⊥ )) = (b →4 c)
13	10, 12	cbtr 182	1 a C (b →4 c)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u4lemc5  699