Theorem wran 369

Description: Weak orthomodular law. (Contributed by NM, 24-Sep-1997.)

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 40	. . 3 (a ∩ c) = (a⊥ ∪ c⊥ )⊥
2		df-a 40	. . 3 (b ∩ c) = (b⊥ ∪ c⊥ )⊥
3	1, 2	2bi 99	. 2 ((a ∩ c) ≡ (b ∩ c)) = ((a⊥ ∪ c⊥ )⊥ ≡ (b⊥ ∪ c⊥ )⊥ )
4		wran.1	. . . . 5 (a ≡ b) = 1
5	4	wr4 199	. . . 4 (a⊥ ≡ b⊥ ) = 1
6	5	wr5-2v 366	. . 3 ((a⊥ ∪ c⊥ ) ≡ (b⊥ ∪ c⊥ )) = 1
7	6	wr4 199	. 2 ((a⊥ ∪ c⊥ )⊥ ≡ (b⊥ ∪ c⊥ )⊥ ) = 1
8	3, 7	ax-r2 36	1 ((a ∩ c) ≡ (b ∩ c)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  wlan  370  wr2  371  w2an  373  wcomlem  382  wlel  392  wleran  394  wbctr  410  wcom3i  422  wfh2  424