Theorem u4lemc2 671

Description: Commutation theorem for non-tollens implication.

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 466	. . . 4 a C (b ^ c)
4	1	comcom2 175	. . . . 5 a C b_|_
5	4, 2	com2an 466	. . . 4 a C (b_|_ ^ c)
6	3, 5	com2or 465	. . 3 a C ((b ^ c) v (b_|_ ^ c))
7	4, 2	com2or 465	. . . 4 a C (b_|_ v c)
8	2	comcom2 175	. . . 4 a C c_|_
9	7, 8	com2an 466	. . 3 a C ((b_|_ v c) ^ c_|_)
10	6, 9	com2or 465	. 2 a C (((b ^ c) v (b_|_ ^ c)) v ((b_|_ v c) ^ c_|_))
11		df-i4 46	. . 3 (b ->4 c) = (((b ^ c) v (b_|_ ^ c)) v ((b_|_ v c) ^ c_|_))
12	11	ax-r1 34	. 2 (((b ^ c) v (b_|_ ^ c)) v ((b_|_ v c) ^ c_|_)) = (b ->4 c)
13	10, 12	cbtr 174	1 a C (b ->4 c)

Syntax hints:   C wc 3  _|_wn 4   v wo 6   ^ wa 7   ->4 wi4 16

This theorem is referenced by:  u4lemc5 681

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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