Theorem oa3-4lem 963

Description: Lemma for 3-OA(4). Equivalence with substitution into 6-OA dual.

Assertion

Ref	Expression
oa3-4lem	(a_|_ ^ (a v (b ^ (((a ^ b) v (a_|_ ^ b_|_)) v (((a ^ c) v (a_|_ ^ 1)) ^ ((b ^ c) v (b_|_ ^ 1))))))) = (a_|_ ^ (a v (b ^ ((a == b) v ((a ->1 c) ^ (b ->1 c))))))

Proof of Theorem oa3-4lem

Step	Hyp	Ref	Expression
1		dfb 86	. . . . . 6 (a == b) = ((a ^ b) v (a_|_ ^ b_|_))
2		ax-a2 30	. . . . . . . 8 (a_|_ v (a ^ c)) = ((a ^ c) v a_|_)
3		df-i1 43	. . . . . . . 8 (a ->1 c) = (a_|_ v (a ^ c))
4		an1 98	. . . . . . . . 9 (a_|_ ^ 1) = a_|_
5	4	lor 66	. . . . . . . 8 ((a ^ c) v (a_|_ ^ 1)) = ((a ^ c) v a_|_)
6	2, 3, 5	3tr1 60	. . . . . . 7 (a ->1 c) = ((a ^ c) v (a_|_ ^ 1))
7		ax-a2 30	. . . . . . . 8 (b_|_ v (b ^ c)) = ((b ^ c) v b_|_)
8		df-i1 43	. . . . . . . 8 (b ->1 c) = (b_|_ v (b ^ c))
9		an1 98	. . . . . . . . 9 (b_|_ ^ 1) = b_|_
10	9	lor 66	. . . . . . . 8 ((b ^ c) v (b_|_ ^ 1)) = ((b ^ c) v b_|_)
11	7, 8, 10	3tr1 60	. . . . . . 7 (b ->1 c) = ((b ^ c) v (b_|_ ^ 1))
12	6, 11	2an 72	. . . . . 6 ((a ->1 c) ^ (b ->1 c)) = (((a ^ c) v (a_|_ ^ 1)) ^ ((b ^ c) v (b_|_ ^ 1)))
13	1, 12	2or 67	. . . . 5 ((a == b) v ((a ->1 c) ^ (b ->1 c))) = (((a ^ b) v (a_|_ ^ b_|_)) v (((a ^ c) v (a_|_ ^ 1)) ^ ((b ^ c) v (b_|_ ^ 1))))
14	13	ax-r1 34	. . . 4 (((a ^ b) v (a_|_ ^ b_|_)) v (((a ^ c) v (a_|_ ^ 1)) ^ ((b ^ c) v (b_|_ ^ 1)))) = ((a == b) v ((a ->1 c) ^ (b ->1 c)))
15	14	lan 70	. . 3 (b ^ (((a ^ b) v (a_|_ ^ b_|_)) v (((a ^ c) v (a_|_ ^ 1)) ^ ((b ^ c) v (b_|_ ^ 1))))) = (b ^ ((a == b) v ((a ->1 c) ^ (b ->1 c))))
16	15	lor 66	. 2 (a v (b ^ (((a ^ b) v (a_|_ ^ b_|_)) v (((a ^ c) v (a_|_ ^ 1)) ^ ((b ^ c) v (b_|_ ^ 1)))))) = (a v (b ^ ((a == b) v ((a ->1 c) ^ (b ->1 c)))))
17	16	lan 70	1 (a_|_ ^ (a v (b ^ (((a ^ b) v (a_|_ ^ b_|_)) v (((a ^ c) v (a_|_ ^ 1)) ^ ((b ^ c) v (b_|_ ^ 1))))))) = (a_|_ ^ (a v (b ^ ((a == b) v ((a ->1 c) ^ (b ->1 c))))))

Syntax hints:   = wb 1  _|_wn 4   == tb 5   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-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

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

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