Theorem com2i3 509

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

Hypotheses

Ref	Expression
com2i3.1	a C b
com2i3.2	a C c

Assertion

Ref	Expression
com2i3	a C (b →3 c)

Proof of Theorem com2i3

Step	Hyp	Ref	Expression
1		com2i3.1	. . . . . 6 a C b
2	1	comcom2 183	. . . . 5 a C b⊥
3		com2i3.2	. . . . 5 a C c
4	2, 3	com2an 484	. . . 4 a C (b⊥ ∩ c)
5	3	comcom2 183	. . . . 5 a C c⊥
6	2, 5	com2an 484	. . . 4 a C (b⊥ ∩ c⊥ )
7	4, 6	com2or 483	. . 3 a C ((b⊥ ∩ c) ∪ (b⊥ ∩ c⊥ ))
8	2, 3	com2or 483	. . . 4 a C (b⊥ ∪ c)
9	1, 8	com2an 484	. . 3 a C (b ∩ (b⊥ ∪ c))
10	7, 9	com2or 483	. 2 a C (((b⊥ ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ (b ∩ (b⊥ ∪ c)))
11		df-i3 46	. . 3 (b →3 c) = (((b⊥ ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ (b ∩ (b⊥ ∪ c)))
12	11	ax-r1 35	. 2 (((b⊥ ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ (b ∩ (b⊥ ∪ c))) = (b →3 c)
13	10, 12	cbtr 182	1 a C (b →3 c)

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:  comi32  510  u3lemc2  688