Theorem oa3-2lemb 959

Description: Lemma for 3-OA(2). Equivalence with substitution into 4-OA.

Assertion

Ref	Expression
oa3-2lemb	((a ->1 c) ^ (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c)))))))) = ((a ->1 c) ^ (a v (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))))

Proof of Theorem oa3-2lemb

Step	Hyp	Ref	Expression
1		ax-a3 31	. . . . 5 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))) = ((a ^ b) v (((a ->1 c) ^ (b ->1 c)) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))))
2		i1id 267	. . . . . . . . . . . . 13 (c ->1 c) = 1
3	2	lan 70	. . . . . . . . . . . 12 ((a ->1 c) ^ (c ->1 c)) = ((a ->1 c) ^ 1)
4		an1 98	. . . . . . . . . . . 12 ((a ->1 c) ^ 1) = (a ->1 c)
5	3, 4	ax-r2 35	. . . . . . . . . . 11 ((a ->1 c) ^ (c ->1 c)) = (a ->1 c)
6	5	lor 66	. . . . . . . . . 10 ((a ^ c) v ((a ->1 c) ^ (c ->1 c))) = ((a ^ c) v (a ->1 c))
7		or12 73	. . . . . . . . . . . 12 ((a ^ c) v (a_|_ v (a ^ c))) = (a_|_ v ((a ^ c) v (a ^ c)))
8		oridm 102	. . . . . . . . . . . . 13 ((a ^ c) v (a ^ c)) = (a ^ c)
9	8	lor 66	. . . . . . . . . . . 12 (a_|_ v ((a ^ c) v (a ^ c))) = (a_|_ v (a ^ c))
10	7, 9	ax-r2 35	. . . . . . . . . . 11 ((a ^ c) v (a_|_ v (a ^ c))) = (a_|_ v (a ^ c))
11		df-i1 43	. . . . . . . . . . . 12 (a ->1 c) = (a_|_ v (a ^ c))
12	11	lor 66	. . . . . . . . . . 11 ((a ^ c) v (a ->1 c)) = ((a ^ c) v (a_|_ v (a ^ c)))
13	10, 12, 11	3tr1 60	. . . . . . . . . 10 ((a ^ c) v (a ->1 c)) = (a ->1 c)
14	6, 13	ax-r2 35	. . . . . . . . 9 ((a ^ c) v ((a ->1 c) ^ (c ->1 c))) = (a ->1 c)
15	2	lan 70	. . . . . . . . . . . 12 ((b ->1 c) ^ (c ->1 c)) = ((b ->1 c) ^ 1)
16		an1 98	. . . . . . . . . . . 12 ((b ->1 c) ^ 1) = (b ->1 c)
17	15, 16	ax-r2 35	. . . . . . . . . . 11 ((b ->1 c) ^ (c ->1 c)) = (b ->1 c)
18	17	lor 66	. . . . . . . . . 10 ((b ^ c) v ((b ->1 c) ^ (c ->1 c))) = ((b ^ c) v (b ->1 c))
19		or12 73	. . . . . . . . . . . 12 ((b ^ c) v (b_|_ v (b ^ c))) = (b_|_ v ((b ^ c) v (b ^ c)))
20		oridm 102	. . . . . . . . . . . . 13 ((b ^ c) v (b ^ c)) = (b ^ c)
21	20	lor 66	. . . . . . . . . . . 12 (b_|_ v ((b ^ c) v (b ^ c))) = (b_|_ v (b ^ c))
22	19, 21	ax-r2 35	. . . . . . . . . . 11 ((b ^ c) v (b_|_ v (b ^ c))) = (b_|_ v (b ^ c))
23		df-i1 43	. . . . . . . . . . . 12 (b ->1 c) = (b_|_ v (b ^ c))
24	23	lor 66	. . . . . . . . . . 11 ((b ^ c) v (b ->1 c)) = ((b ^ c) v (b_|_ v (b ^ c)))
25	22, 24, 23	3tr1 60	. . . . . . . . . 10 ((b ^ c) v (b ->1 c)) = (b ->1 c)
26	18, 25	ax-r2 35	. . . . . . . . 9 ((b ^ c) v ((b ->1 c) ^ (c ->1 c))) = (b ->1 c)
27	14, 26	2an 72	. . . . . . . 8 (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c)))) = ((a ->1 c) ^ (b ->1 c))
28	27	lor 66	. . . . . . 7 (((a ->1 c) ^ (b ->1 c)) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))) = (((a ->1 c) ^ (b ->1 c)) v ((a ->1 c) ^ (b ->1 c)))
29		oridm 102	. . . . . . 7 (((a ->1 c) ^ (b ->1 c)) v ((a ->1 c) ^ (b ->1 c))) = ((a ->1 c) ^ (b ->1 c))
30	28, 29	ax-r2 35	. . . . . 6 (((a ->1 c) ^ (b ->1 c)) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))) = ((a ->1 c) ^ (b ->1 c))
31	30	lor 66	. . . . 5 ((a ^ b) v (((a ->1 c) ^ (b ->1 c)) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c)))))) = ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
32	1, 31	ax-r2 35	. . . 4 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))) = ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
33	32	lan 70	. . 3 (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c)))))) = (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))
34	33	lor 66	. 2 (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c))))))) = (a v (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))))
35	34	lan 70	1 ((a ->1 c) ^ (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ c) v ((a ->1 c) ^ (c ->1 c))) ^ ((b ^ c) v ((b ->1 c) ^ (c ->1 c)))))))) = ((a ->1 c) ^ (a v (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13

This theorem is referenced by:  oa3-2to4 968

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

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

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