Theorem wr5 413

Description: Proof of weak orthomodular law from weaker-looking equivalent, wom3 349, which in turn is derived from ax-wom 343.

Hypothesis

Ref	Expression
wr5.1	(a ≡ b) = 1

Assertion

Ref	Expression
wr5	((a ∪ c) ≡ (b ∪ c)) = 1

Proof of Theorem wr5

Step	Hyp	Ref	Expression
1		wr5.1	. . . . . . 7 (a ≡ b) = 1
2	1	wom3 349	. . . . . 6 a ≤ ((a ∪ c) ≡ (b ∪ c))
3		bicom 88	. . . . . . . . 9 (b ≡ a) = (a ≡ b)
4	3, 1	ax-r2 35	. . . . . . . 8 (b ≡ a) = 1
5	4	wom3 349	. . . . . . 7 b ≤ ((b ∪ c) ≡ (a ∪ c))
6		bicom 88	. . . . . . 7 ((b ∪ c) ≡ (a ∪ c)) = ((a ∪ c) ≡ (b ∪ c))
7	5, 6	lbtr 131	. . . . . 6 b ≤ ((a ∪ c) ≡ (b ∪ c))
8	2, 7	le2or 160	. . . . 5 (a ∪ b) ≤ (((a ∪ c) ≡ (b ∪ c)) ∪ ((a ∪ c) ≡ (b ∪ c)))
9		oridm 102	. . . . 5 (((a ∪ c) ≡ (b ∪ c)) ∪ ((a ∪ c) ≡ (b ∪ c))) = ((a ∪ c) ≡ (b ∪ c))
10	8, 9	lbtr 131	. . . 4 (a ∪ b) ≤ ((a ∪ c) ≡ (b ∪ c))
11	10	df-le2 123	. . 3 ((a ∪ b) ∪ ((a ∪ c) ≡ (b ∪ c))) = ((a ∪ c) ≡ (b ∪ c))
12	11	ax-r1 34	. 2 ((a ∪ c) ≡ (b ∪ c)) = ((a ∪ b) ∪ ((a ∪ c) ≡ (b ∪ c)))
13		ka4lemo 220	. 2 ((a ∪ b) ∪ ((a ∪ c) ≡ (b ∪ c))) = 1
14	12, 13	ax-r2 35	1 ((a ∪ c) ≡ (b ∪ c)) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6  1wt 9

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

metamath.org