Theorem oa3-1to5 973

Description: Derivation of an equivalent of the second "universal" 3-OA U2 from an equivalent of the first "universal" 3-OA U1. This shows that U2 is redundant in a system containg U1. The hypothesis is theorem oal1 980.

Hypothesis

Ref	Expression
oa3-1to5.1	((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))) =< (b ->1 c)

Assertion

Ref	Expression
oa3-1to5	(c ^ ((b ->1 c) v ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))))) =< (b_|_ ->1 c)

Proof of Theorem oa3-1to5

Step	Hyp	Ref	Expression
1		leid 140	. . . . 5 (b ->1 c) =< (b ->1 c)
2		oa3-1to5.1	. . . . 5 ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))) =< (b ->1 c)
3	1, 2	lel2or 162	. . . 4 ((b ->1 c) v ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c))))) =< (b ->1 c)
4	3	lelan 159	. . 3 (c ^ ((b ->1 c) v ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))))) =< (c ^ (b ->1 c))
5		ax-a1 29	. . . . . . . 8 b = b_|__|_
6	5	ran 71	. . . . . . 7 (b ^ c) = (b_|__|_ ^ c)
7	6	ax-r5 37	. . . . . 6 ((b ^ c) v (b_|_ ^ c)) = ((b_|__|_ ^ c) v (b_|_ ^ c))
8		ax-a2 30	. . . . . 6 ((b_|__|_ ^ c) v (b_|_ ^ c)) = ((b_|_ ^ c) v (b_|__|_ ^ c))
9	7, 8	ax-r2 35	. . . . 5 ((b ^ c) v (b_|_ ^ c)) = ((b_|_ ^ c) v (b_|__|_ ^ c))
10		u1lemab 592	. . . . 5 ((b ->1 c) ^ c) = ((b ^ c) v (b_|_ ^ c))
11		u1lemab 592	. . . . 5 ((b_|_ ->1 c) ^ c) = ((b_|_ ^ c) v (b_|__|_ ^ c))
12	9, 10, 11	3tr1 60	. . . 4 ((b ->1 c) ^ c) = ((b_|_ ->1 c) ^ c)
13		ancom 68	. . . 4 (c ^ (b ->1 c)) = ((b ->1 c) ^ c)
14		ancom 68	. . . 4 (c ^ (b_|_ ->1 c)) = ((b_|_ ->1 c) ^ c)
15	12, 13, 14	3tr1 60	. . 3 (c ^ (b ->1 c)) = (c ^ (b_|_ ->1 c))
16	4, 15	lbtr 131	. 2 (c ^ ((b ->1 c) v ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))))) =< (c ^ (b_|_ ->1 c))
17		lear 153	. 2 (c ^ (b_|_ ->1 c)) =< (b_|_ ->1 c)
18	16, 17	letr 129	1 (c ^ ((b ->1 c) v ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))))) =< (b_|_ ->1 c)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13

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

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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