Theorem nomb41 299

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

Assertion

Ref	Expression
nomb41	(a ≡4 b) = (b ≡1 a)

Proof of Theorem nomb41

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 (b⊥ ∪ (a ∩ b)) = (b⊥ ∪ (b ∩ a))
4	1, 3	2an 79	. 2 ((a⊥ ∪ b) ∩ (b⊥ ∪ (a ∩ b))) = ((b ∪ a⊥ ) ∩ (b⊥ ∪ (b ∩ a)))
5		df-id4 53	. 2 (a ≡4 b) = ((a⊥ ∪ b) ∩ (b⊥ ∪ (a ∩ b)))
6		df-id1 50	. 2 (b ≡1 a) = ((b ∪ a⊥ ) ∩ (b⊥ ∪ (b ∩ a)))
7	4, 5, 6	3tr1 63	1 (a ≡4 b) = (b ≡1 a)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   ≡₁ wid1 18   ≡₄ wid4 21

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-id1 50  df-id4 53

This theorem is used by:  nomcon3  304  nomcon4  305  nom31  320  nom34  323  nom61  338  nom64  341