Theorem 4oath1 1020

Description: Proper 4-OA theorem.

Hypotheses

Ref	Expression
4oa.1	e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
4oa.2	f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)

Assertion

Ref	Expression
4oath1	((a ->1 d) ^ f) = ((a ->1 d) ^ (b ->1 d))

Proof of Theorem 4oath1

Step	Hyp	Ref	Expression
1		4oa.1	. . . . . 6 e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
2		4oa.2	. . . . . 6 f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)
3	1, 2	4oaiii 1019	. . . . 5 ((a ->1 d) ^ f) = ((b ->1 d) ^ f)
4	3	lan 70	. . . 4 (((a ->1 d) ^ f) ^ ((a ->1 d) ^ f)) = (((a ->1 d) ^ f) ^ ((b ->1 d) ^ f))
5		or32 75	. . . . . . 7 (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e) = (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
6	2, 5	ax-r2 35	. . . . . 6 f = (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
7	6	lan 70	. . . . 5 ((a ->1 d) ^ f) = ((a ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))))
8	6	lan 70	. . . . 5 ((b ->1 d) ^ f) = ((b ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))))
9	7, 8	2an 72	. . . 4 (((a ->1 d) ^ f) ^ ((b ->1 d) ^ f)) = (((a ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))) ^ ((b ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))))
10	4, 9	ax-r2 35	. . 3 (((a ->1 d) ^ f) ^ ((a ->1 d) ^ f)) = (((a ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))) ^ ((b ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))))
11		anidm 103	. . . 4 (((a ->1 d) ^ f) ^ ((a ->1 d) ^ f)) = ((a ->1 d) ^ f)
12	11	ax-r1 34	. . 3 ((a ->1 d) ^ f) = (((a ->1 d) ^ f) ^ ((a ->1 d) ^ f))
13		anandir 107	. . 3 (((a ->1 d) ^ (b ->1 d)) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))) = (((a ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))) ^ ((b ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))))
14	10, 12, 13	3tr1 60	. 2 ((a ->1 d) ^ f) = (((a ->1 d) ^ (b ->1 d)) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))))
15		ax-a2 30	. . 3 (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))) = (((a ->1 d) ^ (b ->1 d)) v ((a ^ b) v e))
16	15	lan 70	. 2 (((a ->1 d) ^ (b ->1 d)) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))) = (((a ->1 d) ^ (b ->1 d)) ^ (((a ->1 d) ^ (b ->1 d)) v ((a ^ b) v e)))
17		a5c 113	. 2 (((a ->1 d) ^ (b ->1 d)) ^ (((a ->1 d) ^ (b ->1 d)) v ((a ^ b) v e))) = ((a ->1 d) ^ (b ->1 d))
18	14, 16, 17	3tr 62	1 ((a ->1 d) ^ f) = ((a ->1 d) ^ (b ->1 d))

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

This theorem is referenced by:  4oagen1 1021

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  ax-4oa 1012

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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