Theorem u4lem3 734

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u4lem3	(a ->4 (b ->4 a)) = (a_|_ v ((a ^ b) v (a ^ b_|_)))

Proof of Theorem u4lem3

Step	Hyp	Ref	Expression
1		u4lemc1 665	. . 3 a C (b ->4 a)
2	1	u4lemc4 686	. 2 (a ->4 (b ->4 a)) = (a_|_ v (b ->4 a))
3		ax-a2 30	. . 3 (a_|_ v (b ->4 a)) = ((b ->4 a) v a_|_)
4		u4lemonb 620	. . . 4 ((b ->4 a) v a_|_) = (((b ^ a) v (b_|_ ^ a)) v a_|_)
5		ancom 68	. . . . . . 7 (b ^ a) = (a ^ b)
6		ancom 68	. . . . . . 7 (b_|_ ^ a) = (a ^ b_|_)
7	5, 6	2or 67	. . . . . 6 ((b ^ a) v (b_|_ ^ a)) = ((a ^ b) v (a ^ b_|_))
8	7	ax-r5 37	. . . . 5 (((b ^ a) v (b_|_ ^ a)) v a_|_) = (((a ^ b) v (a ^ b_|_)) v a_|_)
9		ax-a2 30	. . . . 5 (((a ^ b) v (a ^ b_|_)) v a_|_) = (a_|_ v ((a ^ b) v (a ^ b_|_)))
10	8, 9	ax-r2 35	. . . 4 (((b ^ a) v (b_|_ ^ a)) v a_|_) = (a_|_ v ((a ^ b) v (a ^ b_|_)))
11	4, 10	ax-r2 35	. . 3 ((b ->4 a) v a_|_) = (a_|_ v ((a ^ b) v (a ^ b_|_)))
12	3, 11	ax-r2 35	. 2 (a_|_ v (b ->4 a)) = (a_|_ v ((a ^ b) v (a ^ b_|_)))
13	2, 12	ax-r2 35	1 (a ->4 (b ->4 a)) = (a_|_ v ((a ^ b) v (a ^ b_|_)))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->4 wi4 16

This theorem is referenced by:  u4lem3n 737  u4lem4 741

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-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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