Theorem oi3oa3 715

Description: An attempt at the OA3 conjecture, which is true if (a == b) = 1. (Contributed by Josiah Burroughs 27-May-04.)

Hypothesis

Ref	Expression
oi3oa3.1	1 = (b == a)

Assertion

Ref	Expression
oi3oa3	(((a ->1 c) ^ (b ->1 c)) v ((((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) ->1 c) ^ (((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) ->1 c))) = 1

Proof of Theorem oi3oa3

Step	Hyp	Ref	Expression
1		oi3oa3.1	. . . . . . . 8 1 = (b == a)
2	1	oi3oa3lem1 714	. . . . . . 7 (((a ->1 c) ^ (b ->1 c)) v (a ^ b)) = 1
3	2	lan 70	. . . . . 6 ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) = ((a ->1 c) ^ 1)
4		an1 98	. . . . . 6 ((a ->1 c) ^ 1) = (a ->1 c)
5	3, 4	ax-r2 35	. . . . 5 ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) = (a ->1 c)
6	5	ud1lem0b 248	. . . 4 (((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) ->1 c) = ((a ->1 c) ->1 c)
7	2	lan 70	. . . . . 6 ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) = ((b ->1 c) ^ 1)
8		an1 98	. . . . . 6 ((b ->1 c) ^ 1) = (b ->1 c)
9	7, 8	ax-r2 35	. . . . 5 ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) = (b ->1 c)
10	9	ud1lem0b 248	. . . 4 (((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) ->1 c) = ((b ->1 c) ->1 c)
11	6, 10	2an 72	. . 3 ((((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) ->1 c) ^ (((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) ->1 c)) = (((a ->1 c) ->1 c) ^ ((b ->1 c) ->1 c))
12	11	lor 66	. 2 (((a ->1 c) ^ (b ->1 c)) v ((((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) ->1 c) ^ (((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) ->1 c))) = (((a ->1 c) ^ (b ->1 c)) v (((a ->1 c) ->1 c) ^ ((b ->1 c) ->1 c)))
13		ax-a2 30	. 2 (((a ->1 c) ^ (b ->1 c)) v (((a ->1 c) ->1 c) ^ ((b ->1 c) ->1 c))) = ((((a ->1 c) ->1 c) ^ ((b ->1 c) ->1 c)) v ((a ->1 c) ^ (b ->1 c)))
14	1	r3a 422	. . . . 5 b = a
15	14	ud1lem0b 248	. . . 4 (b ->1 c) = (a ->1 c)
16	15	1bi 111	. . 3 1 = ((b ->1 c) == (a ->1 c))
17	16	oi3oa3lem1 714	. 2 ((((a ->1 c) ->1 c) ^ ((b ->1 c) ->1 c)) v ((a ->1 c) ^ (b ->1 c))) = 1
18	12, 13, 17	3tr 62	1 (((a ->1 c) ^ (b ->1 c)) v ((((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) ->1 c) ^ (((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v (a ^ b))) ->1 c))) = 1

Syntax hints:   = wb 1   == tb 5   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13

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-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43

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