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 v c) == (b v c)) = 1

Proof of Theorem wr5

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

Syntax hints:   = wb 1   == tb 5   v 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

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