Theorem oa3-2lema 958

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

Assertion

Ref	Expression
oa3-2lema	((a ->1 c) ^ (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c)))))))) = ((a ->1 c) ^ (a v (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))))

Proof of Theorem oa3-2lema

Step	Hyp	Ref	Expression
1		ax-a3 31	. . . . 5 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c))))) = ((a ^ b) v (((a ->1 c) ^ (b ->1 c)) v (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c))))))
2		an0 100	. . . . . . . . . . 11 (a ^ 0) = 0
3	2	ax-r5 37	. . . . . . . . . 10 ((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) = (0 v ((a ->1 c) ^ (0 ->1 c)))
4		ax-a2 30	. . . . . . . . . 10 (0 v ((a ->1 c) ^ (0 ->1 c))) = (((a ->1 c) ^ (0 ->1 c)) v 0)
5		or0 94	. . . . . . . . . . 11 (((a ->1 c) ^ (0 ->1 c)) v 0) = ((a ->1 c) ^ (0 ->1 c))
6		0i1 265	. . . . . . . . . . . 12 (0 ->1 c) = 1
7	6	lan 70	. . . . . . . . . . 11 ((a ->1 c) ^ (0 ->1 c)) = ((a ->1 c) ^ 1)
8		an1 98	. . . . . . . . . . 11 ((a ->1 c) ^ 1) = (a ->1 c)
9	5, 7, 8	3tr 62	. . . . . . . . . 10 (((a ->1 c) ^ (0 ->1 c)) v 0) = (a ->1 c)
10	3, 4, 9	3tr 62	. . . . . . . . 9 ((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) = (a ->1 c)
11		an0 100	. . . . . . . . . . 11 (b ^ 0) = 0
12	11	ax-r5 37	. . . . . . . . . 10 ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c))) = (0 v ((b ->1 c) ^ (0 ->1 c)))
13		ax-a2 30	. . . . . . . . . 10 (0 v ((b ->1 c) ^ (0 ->1 c))) = (((b ->1 c) ^ (0 ->1 c)) v 0)
14		or0 94	. . . . . . . . . . 11 (((b ->1 c) ^ (0 ->1 c)) v 0) = ((b ->1 c) ^ (0 ->1 c))
15	6	lan 70	. . . . . . . . . . 11 ((b ->1 c) ^ (0 ->1 c)) = ((b ->1 c) ^ 1)
16		an1 98	. . . . . . . . . . 11 ((b ->1 c) ^ 1) = (b ->1 c)
17	14, 15, 16	3tr 62	. . . . . . . . . 10 (((b ->1 c) ^ (0 ->1 c)) v 0) = (b ->1 c)
18	12, 13, 17	3tr 62	. . . . . . . . 9 ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c))) = (b ->1 c)
19	10, 18	2an 72	. . . . . . . 8 (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c)))) = ((a ->1 c) ^ (b ->1 c))
20	19	lor 66	. . . . . . 7 (((a ->1 c) ^ (b ->1 c)) v (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c))))) = (((a ->1 c) ^ (b ->1 c)) v ((a ->1 c) ^ (b ->1 c)))
21		oridm 102	. . . . . . 7 (((a ->1 c) ^ (b ->1 c)) v ((a ->1 c) ^ (b ->1 c))) = ((a ->1 c) ^ (b ->1 c))
22	20, 21	ax-r2 35	. . . . . 6 (((a ->1 c) ^ (b ->1 c)) v (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c))))) = ((a ->1 c) ^ (b ->1 c))
23	22	lor 66	. . . . 5 ((a ^ b) v (((a ->1 c) ^ (b ->1 c)) v (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c)))))) = ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
24	1, 23	ax-r2 35	. . . 4 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c))))) = ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
25	24	lan 70	. . 3 (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c)))))) = (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))
26	25	lor 66	. 2 (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c))))))) = (a v (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))))
27	26	lan 70	1 ((a ->1 c) ^ (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ 0) v ((a ->1 c) ^ (0 ->1 c))) ^ ((b ^ 0) v ((b ->1 c) ^ (0 ->1 c)))))))) = ((a ->1 c) ^ (a v (b ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))))

Syntax hints:   = wb 1   v wo 6   ^ wa 7  1wt 9  0wf 10   ->1 wi1 13

This theorem is referenced by:  oa3-2to2s 970

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

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

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