Theorem comi31 490

Description: Commutation theorem.

Assertion

Ref	Expression
comi31	a C (a →3 b)

Proof of Theorem comi31

Step	Hyp	Ref	Expression
1		coman1 177	. . . . . . 7 (a⊥ ∩ b) C a⊥
2	1	comcom 435	. . . . . 6 a⊥ C (a⊥ ∩ b)
3	2	comcom2 175	. . . . 5 a⊥ C (a⊥ ∩ b)⊥
4	3	comcom5 440	. . . 4 a C (a⊥ ∩ b)
5		coman1 177	. . . . . . 7 (a⊥ ∩ b⊥ ) C a⊥
6	5	comcom 435	. . . . . 6 a⊥ C (a⊥ ∩ b⊥ )
7	6	comcom2 175	. . . . 5 a⊥ C (a⊥ ∩ b⊥ )⊥
8	7	comcom5 440	. . . 4 a C (a⊥ ∩ b⊥ )
9	4, 8	com2or 465	. . 3 a C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
10		coman1 177	. . . 4 (a ∩ (a⊥ ∪ b)) C a
11	10	comcom 435	. . 3 a C (a ∩ (a⊥ ∪ b))
12	9, 11	com2or 465	. 2 a C (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
13		df-i3 45	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
14	13	ax-r1 34	. 2 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (a →3 b)
15	12, 14	cbtr 174	1 a C (a →3 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 15

This theorem is referenced by:  i3abs3 506  u3lemc1 664  u3lemc5 680  u3lem1 718  u3lem2 728  u3lem5 745  u3lem6 749  u3lem7 756  u3lem8 765  u3lem9 766  u3lem13a 771  u3lem13b 772

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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