Theorem wcom2an 428

Description: Thm. 4.2 Beran p. 49. (Contributed by NM, 10-Nov-1998.)

Hypotheses

Ref	Expression
wfh.1	C (a, b) = 1
wfh.2	C (a, c) = 1

Assertion

Ref	Expression
wcom2an	C (a, (b ∩ c)) = 1

Proof of Theorem wcom2an

Step	Hyp	Ref	Expression
1		wfh.1	. . . . 5 C (a, b) = 1
2	1	wcomcom4 417	. . . 4 C (a⊥ , b⊥ ) = 1
3		wfh.2	. . . . 5 C (a, c) = 1
4	3	wcomcom4 417	. . . 4 C (a⊥ , c⊥ ) = 1
5	2, 4	wcom2or 427	. . 3 C (a⊥ , (b⊥ ∪ c⊥ )) = 1
6		df-a 40	. . . . . 6 (b ∩ c) = (b⊥ ∪ c⊥ )⊥
7	6	con2 67	. . . . 5 (b ∩ c)⊥ = (b⊥ ∪ c⊥ )
8	7	ax-r1 35	. . . 4 (b⊥ ∪ c⊥ ) = (b ∩ c)⊥
9	8	bi1 118	. . 3 ((b⊥ ∪ c⊥ ) ≡ (b ∩ c)⊥ ) = 1
10	5, 9	wcbtr 411	. 2 C (a⊥ , (b ∩ c)⊥ ) = 1
11	10	wcomcom5 420	1 C (a, (b ∩ c)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29

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-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134

This theorem is referenced by:  ska4  433