Theorem u4lemc1 665

Description: Commutation theorem for non-tollens implication.

Assertion

Ref	Expression
u4lemc1	b C (a →4 b)

Proof of Theorem u4lemc1

Step	Hyp	Ref	Expression
1		comanr2 447	. . . 4 b C (a ∩ b)
2		comanr2 447	. . . 4 b C (a⊥ ∩ b)
3	1, 2	com2or 465	. . 3 b C ((a ∩ b) ∪ (a⊥ ∩ b))
4		comorr2 445	. . . 4 b C (a⊥ ∪ b)
5		comid 179	. . . . 5 b C b
6	5	comcom2 175	. . . 4 b C b⊥
7	4, 6	com2an 466	. . 3 b C ((a⊥ ∪ b) ∩ b⊥ )
8	3, 7	com2or 465	. 2 b C (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
9		df-i4 46	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
10	9	ax-r1 34	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a →4 b)
11	8, 10	cbtr 174	1 b C (a →4 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 16

This theorem is referenced by:  u4lemc3 676  u4lem2 729  u4lem3 734  u24lem 752

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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