Theorem oa3-5lem 964

Description: Lemma for 3-OA(5). Equivalence with substitution into 6-OA dual.

Assertion

Ref	Expression
oa3-5lem	((a ->1 c) ^ (a v (c ^ (((a ^ c) v ((a ->1 c) ^ 1)) v (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ b) v (1 ^ (b ->1 c)))))))) = ((a ->1 c) ^ (a v (c ^ ((a ->1 c) v ((b ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))))))

Proof of Theorem oa3-5lem

Step	Hyp	Ref	Expression
1		or12 73	. . . . . . 7 ((a ^ c) v (a_|_ v (a ^ c))) = (a_|_ v ((a ^ c) v (a ^ c)))
2		oridm 102	. . . . . . . 8 ((a ^ c) v (a ^ c)) = (a ^ c)
3	2	lor 66	. . . . . . 7 (a_|_ v ((a ^ c) v (a ^ c))) = (a_|_ v (a ^ c))
4	1, 3	ax-r2 35	. . . . . 6 ((a ^ c) v (a_|_ v (a ^ c))) = (a_|_ v (a ^ c))
5		an1 98	. . . . . . . 8 ((a ->1 c) ^ 1) = (a ->1 c)
6		df-i1 43	. . . . . . . 8 (a ->1 c) = (a_|_ v (a ^ c))
7	5, 6	ax-r2 35	. . . . . . 7 ((a ->1 c) ^ 1) = (a_|_ v (a ^ c))
8	7	lor 66	. . . . . 6 ((a ^ c) v ((a ->1 c) ^ 1)) = ((a ^ c) v (a_|_ v (a ^ c)))
9	4, 8, 6	3tr1 60	. . . . 5 ((a ^ c) v ((a ->1 c) ^ 1)) = (a ->1 c)
10		or12 73	. . . . . . . . 9 ((c ^ b) v (b_|_ v (b ^ c))) = (b_|_ v ((c ^ b) v (b ^ c)))
11		ancom 68	. . . . . . . . . . . 12 (c ^ b) = (b ^ c)
12	11	ax-r5 37	. . . . . . . . . . 11 ((c ^ b) v (b ^ c)) = ((b ^ c) v (b ^ c))
13		oridm 102	. . . . . . . . . . 11 ((b ^ c) v (b ^ c)) = (b ^ c)
14	12, 13	ax-r2 35	. . . . . . . . . 10 ((c ^ b) v (b ^ c)) = (b ^ c)
15	14	lor 66	. . . . . . . . 9 (b_|_ v ((c ^ b) v (b ^ c))) = (b_|_ v (b ^ c))
16	10, 15	ax-r2 35	. . . . . . . 8 ((c ^ b) v (b_|_ v (b ^ c))) = (b_|_ v (b ^ c))
17		ancom 68	. . . . . . . . . 10 (1 ^ (b ->1 c)) = ((b ->1 c) ^ 1)
18		an1 98	. . . . . . . . . 10 ((b ->1 c) ^ 1) = (b ->1 c)
19		df-i1 43	. . . . . . . . . 10 (b ->1 c) = (b_|_ v (b ^ c))
20	17, 18, 19	3tr 62	. . . . . . . . 9 (1 ^ (b ->1 c)) = (b_|_ v (b ^ c))
21	20	lor 66	. . . . . . . 8 ((c ^ b) v (1 ^ (b ->1 c))) = ((c ^ b) v (b_|_ v (b ^ c)))
22	16, 21, 19	3tr1 60	. . . . . . 7 ((c ^ b) v (1 ^ (b ->1 c))) = (b ->1 c)
23	22	lan 70	. . . . . 6 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ b) v (1 ^ (b ->1 c)))) = (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) ^ (b ->1 c))
24		ancom 68	. . . . . 6 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) ^ (b ->1 c)) = ((b ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))
25	23, 24	ax-r2 35	. . . . 5 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ b) v (1 ^ (b ->1 c)))) = ((b ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))
26	9, 25	2or 67	. . . 4 (((a ^ c) v ((a ->1 c) ^ 1)) v (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ b) v (1 ^ (b ->1 c))))) = ((a ->1 c) v ((b ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))))
27	26	lan 70	. . 3 (c ^ (((a ^ c) v ((a ->1 c) ^ 1)) v (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ b) v (1 ^ (b ->1 c)))))) = (c ^ ((a ->1 c) v ((b ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))))
28	27	lor 66	. 2 (a v (c ^ (((a ^ c) v ((a ->1 c) ^ 1)) v (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ b) v (1 ^ (b ->1 c))))))) = (a v (c ^ ((a ->1 c) v ((b ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))))))
29	28	lan 70	1 ((a ->1 c) ^ (a v (c ^ (((a ^ c) v ((a ->1 c) ^ 1)) v (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ b) v (1 ^ (b ->1 c)))))))) = ((a ->1 c) ^ (a v (c ^ ((a ->1 c) v ((b ->1 c) ^ ((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 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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