Theorem nomb32 292

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

Assertion

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

Proof of Theorem nomb32

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 (a ∪ (a⊥ ∩ b⊥ )) = (a ∪ (b⊥ ∩ a⊥ ))
4	1, 3	2an 72	. 2 ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b⊥ ))) = ((b ∪ a⊥ ) ∩ (a ∪ (b⊥ ∩ a⊥ )))
5		df-id3 51	. 2 (a ≡3 b) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b⊥ )))
6		df-id2 50	. 2 (b ≡2 a) = ((b ∪ a⊥ ) ∩ (a ∪ (b⊥ ∩ a⊥ )))
7	4, 5, 6	3tr1 60	1 (a ≡3 b) = (b ≡2 a)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   ≡₂ wid2 20   ≡₃ wid3 21

This theorem is referenced by:  nomcon3 296  nomcon4 297  nom32 313  nom33 314  nom62 331  nom63 332

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-id2 50  df-id3 51

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