Theorem oal42 915

Description: Derivation of Godowski/Greechie Eq. II from Eq. IV.

Hypothesis

Ref	Expression
oal42.1	(b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< ((b_|_ ^ (a ->2 b)) v (c_|_ ^ (a ->2 c)))

Assertion

Ref	Expression
oal42	(b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< a_|_

Proof of Theorem oal42

Step	Hyp	Ref	Expression
1		oal42.1	. . 3 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< ((b_|_ ^ (a ->2 b)) v (c_|_ ^ (a ->2 c)))
2		ancom 68	. . . . 5 (b_|_ ^ (a ->2 b)) = ((a ->2 b) ^ b_|_)
3		u2lemanb 598	. . . . 5 ((a ->2 b) ^ b_|_) = (a_|_ ^ b_|_)
4	2, 3	ax-r2 35	. . . 4 (b_|_ ^ (a ->2 b)) = (a_|_ ^ b_|_)
5		ancom 68	. . . . 5 (c_|_ ^ (a ->2 c)) = ((a ->2 c) ^ c_|_)
6		u2lemanb 598	. . . . 5 ((a ->2 c) ^ c_|_) = (a_|_ ^ c_|_)
7	5, 6	ax-r2 35	. . . 4 (c_|_ ^ (a ->2 c)) = (a_|_ ^ c_|_)
8	4, 7	2or 67	. . 3 ((b_|_ ^ (a ->2 b)) v (c_|_ ^ (a ->2 c))) = ((a_|_ ^ b_|_) v (a_|_ ^ c_|_))
9	1, 8	lbtr 131	. 2 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< ((a_|_ ^ b_|_) v (a_|_ ^ c_|_))
10		lea 152	. . 3 (a_|_ ^ b_|_) =< a_|_
11		lea 152	. . 3 (a_|_ ^ c_|_) =< a_|_
12	10, 11	lel2or 162	. 2 ((a_|_ ^ b_|_) v (a_|_ ^ c_|_)) =< a_|_
13	9, 12	letr 129	1 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< a_|_

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14

This theorem is referenced by:  oa43v 1008  oa63v 1011

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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