Theorem nomb32 300

Description: Lemma for "Non-Orthomodular Models..." paper. (Contributed by NM, 7-Feb-1999.)

Assertion

Ref	Expression
nomb32	(a ≡3 b) = (b ≡2 a)

Proof of Theorem nomb32

Step	Hyp	Ref	Expression
1		ax-a2 31	. . 3 (a⊥ ∪ b) = (b ∪ a⊥ )
2		ancom 74	. . . 4 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
3	2	lor 70	. . 3 (a ∪ (a⊥ ∩ b⊥ )) = (a ∪ (b⊥ ∩ a⊥ ))
4	1, 3	2an 79	. 2 ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b⊥ ))) = ((b ∪ a⊥ ) ∩ (a ∪ (b⊥ ∩ a⊥ )))
5		df-id3 52	. 2 (a ≡3 b) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b⊥ )))
6		df-id2 51	. 2 (b ≡2 a) = ((b ∪ a⊥ ) ∩ (a ∪ (b⊥ ∩ a⊥ )))
7	4, 5, 6	3tr1 63	1 (a ≡3 b) = (b ≡2 a)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   ≡₂ wid2 19   ≡₃ wid3 20

This proof depends on axioms:  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-id2 51  df-id3 52

This theorem is used by:  nomcon3  304  nomcon4  305  nom32  321  nom33  322  nom62  339  nom63  340