Theorem u4lemc1 683

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

Assertion

Ref	Expression
u4lemc1	b C (a →4 b)

Proof of Theorem u4lemc1

Step	Hyp	Ref	Expression
1		comanr2 465	. . . 4 b C (a ∩ b)
2		comanr2 465	. . . 4 b C (a⊥ ∩ b)
3	1, 2	com2or 483	. . 3 b C ((a ∩ b) ∪ (a⊥ ∩ b))
4		comorr2 463	. . . 4 b C (a⊥ ∪ b)
5		comid 187	. . . . 5 b C b
6	5	comcom2 183	. . . 4 b C b⊥
7	4, 6	com2an 484	. . 3 b C ((a⊥ ∪ b) ∩ b⊥ )
8	3, 7	com2or 483	. 2 b C (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
9		df-i4 47	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
10	9	ax-r1 35	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a →4 b)
11	8, 10	cbtr 182	1 b C (a →4 b)

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:  u4lemc3  694  u4lem2  747  u4lem3  752  u24lem  770