Theorem comi31 508

Description: Commutation theorem. (Contributed by NM, 9-Nov-1997.)

Assertion

Ref	Expression
comi31	a C (a →3 b)

Proof of Theorem comi31

Step	Hyp	Ref	Expression
1		coman1 185	. . . . . . 7 (a⊥ ∩ b) C a⊥
2	1	comcom 453	. . . . . 6 a⊥ C (a⊥ ∩ b)
3	2	comcom2 183	. . . . 5 a⊥ C (a⊥ ∩ b)⊥
4	3	comcom5 458	. . . 4 a C (a⊥ ∩ b)
5		coman1 185	. . . . . . 7 (a⊥ ∩ b⊥ ) C a⊥
6	5	comcom 453	. . . . . 6 a⊥ C (a⊥ ∩ b⊥ )
7	6	comcom2 183	. . . . 5 a⊥ C (a⊥ ∩ b⊥ )⊥
8	7	comcom5 458	. . . 4 a C (a⊥ ∩ b⊥ )
9	4, 8	com2or 483	. . 3 a C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
10		coman1 185	. . . 4 (a ∩ (a⊥ ∪ b)) C a
11	10	comcom 453	. . 3 a C (a ∩ (a⊥ ∪ b))
12	9, 11	com2or 483	. 2 a C (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
13		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
14	13	ax-r1 35	. 2 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (a →3 b)
15	12, 14	cbtr 182	1 a C (a →3 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

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

This theorem is referenced by:  i3abs3  524  u3lemc1  682  u3lemc5  698  u3lem1  736  u3lem2  746  u3lem5  763  u3lem6  767  u3lem7  774  u3lem8  783  u3lem9  784  u3lem13a  789  u3lem13b  790