Theorem wran 351

Description: Weak orthomodular law.

Hypothesis

Ref	Expression
wran.1	(a ≡ b) = 1

Assertion

Ref	Expression
wran	((a ∩ c) ≡ (b ∩ c)) = 1

Proof of Theorem wran

Step	Hyp	Ref	Expression
1		df-a 39	. . 3 (a ∩ c) = (a⊥ ∪ c⊥ )⊥
2		df-a 39	. . 3 (b ∩ c) = (b⊥ ∪ c⊥ )⊥
3	1, 2	2bi 91	. 2 ((a ∩ c) ≡ (b ∩ c)) = ((a⊥ ∪ c⊥ )⊥ ≡ (b⊥ ∪ c⊥ )⊥ )
4		wran.1	. . . . 5 (a ≡ b) = 1
5	4	wr4 191	. . . 4 (a⊥ ≡ b⊥ ) = 1
6	5	wr5-2v 348	. . 3 ((a⊥ ∪ c⊥ ) ≡ (b⊥ ∪ c⊥ )) = 1
7	6	wr4 191	. 2 ((a⊥ ∪ c⊥ )⊥ ≡ (b⊥ ∪ c⊥ )⊥ ) = 1
8	3, 7	ax-r2 35	1 ((a ∩ c) ≡ (b ∩ c)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9

This theorem is referenced by:  wlan 352  wr2 353  w2an 355  wcomlem 364  wlel 374  wleran 376  wbctr 392  wcom3i 404  wfh2 406

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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