Theorem df2c1 468

Description: Dual 'commutes' analogue for ≡ analogue of =. (Contributed by NM, 20-Sep-1998.)

Hypothesis

Ref	Expression
df2c1.1	a = ((a ∪ b) ∩ (a ∪ b⊥ ))

Assertion

Ref	Expression
df2c1	a C b

Proof of Theorem df2c1

Step	Hyp	Ref	Expression
1		df2c1.1	. . . . 5 a = ((a ∪ b) ∩ (a ∪ b⊥ ))
2		df-a 40	. . . . . 6 ((a ∪ b) ∩ (a ∪ b⊥ )) = ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )⊥
3		anor3 90	. . . . . . . . 9 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
4		anor3 90	. . . . . . . . 9 (a⊥ ∩ b⊥ ⊥ ) = (a ∪ b⊥ )⊥
5	3, 4	2or 72	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ )) = ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )
6	5	ax-r1 35	. . . . . . 7 ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ ) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))
7	6	ax-r4 37	. . . . . 6 ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )⊥ = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥
8	2, 7	ax-r2 36	. . . . 5 ((a ∪ b) ∩ (a ∪ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥
9	1, 8	ax-r2 36	. . . 4 a = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥
10	9	con2 67	. . 3 a⊥ = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))
11	10	df-c1 132	. 2 a⊥ C b⊥
12	11	comcom5 458	1 a C b

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  gsth  489