Theorem wlan 370

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

Hypothesis

Ref	Expression
wlan.1	(a ≡ b) = 1

Assertion

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

Proof of Theorem wlan

Step	Hyp	Ref	Expression
1		ancom 74	. . 3 (c ∩ a) = (a ∩ c)
2		ancom 74	. . 3 (c ∩ b) = (b ∩ c)
3	1, 2	2bi 99	. 2 ((c ∩ a) ≡ (c ∩ b)) = ((a ∩ c) ≡ (b ∩ c))
4		wlan.1	. . 3 (a ≡ b) = 1
5	4	wran 369	. 2 ((a ∩ c) ≡ (b ∩ c)) = 1
6	3, 5	ax-r2 36	1 ((c ∩ a) ≡ (c ∩ b)) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∩ 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:  w2an  373  wleoa  376  wom4  380  wcomlem  382  wletr  396  wlbtr  398  wcbtr  411  wcomcom2  415  wcom3ii  419  wfh1  423  wfh2  424  wlem14  430  wdid0id1  1112  wdid0id3  1114  wdid0id4  1115