Theorem nomb41 291

Description: Lemma for "Non-Orthomodular Models..." paper.

Assertion

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

Proof of Theorem nomb41

Step	Hyp	Ref	Expression
1		ax-a2 30	. . 3 (a⊥ ∪ b) = (b ∪ a⊥ )
2		ancom 68	. . . 4 (a ∩ b) = (b ∩ a)
3	2	lor 66	. . 3 (b⊥ ∪ (a ∩ b)) = (b⊥ ∪ (b ∩ a))
4	1, 3	2an 72	. 2 ((a⊥ ∪ b) ∩ (b⊥ ∪ (a ∩ b))) = ((b ∪ a⊥ ) ∩ (b⊥ ∪ (b ∩ a)))
5		df-id4 52	. 2 (a ≡4 b) = ((a⊥ ∪ b) ∩ (b⊥ ∪ (a ∩ b)))
6		df-id1 49	. 2 (b ≡1 a) = ((b ∪ a⊥ ) ∩ (b⊥ ∪ (b ∩ a)))
7	4, 5, 6	3tr1 60	1 (a ≡4 b) = (b ≡1 a)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   ≡₁ wid1 19   ≡₄ wid4 22

This theorem is referenced by:  nomcon3 296  nomcon4 297  nom31 312  nom34 315  nom61 330  nom64 333

This theorem was proved from axioms:  ax-a2 30  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-id1 49  df-id4 52

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