Theorem ud4lem1d 562

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud4lem1d	(((a ->4 b)_|_ v (b ->4 a)) ^ (b ->4 a)_|_) = (((a_|_ v b_|_) ^ (a_|_ v b)) ^ a)

Proof of Theorem ud4lem1d

Step	Hyp	Ref	Expression
1		ud4lem1c 561	. . 3 ((a ->4 b)_|_ v (b ->4 a)) = (a v b_|_)
2		ud4lem0c 272	. . 3 (b ->4 a)_|_ = (((b_|_ v a_|_) ^ (b v a_|_)) ^ ((b ^ a_|_) v a))
3	1, 2	2an 72	. 2 (((a ->4 b)_|_ v (b ->4 a)) ^ (b ->4 a)_|_) = ((a v b_|_) ^ (((b_|_ v a_|_) ^ (b v a_|_)) ^ ((b ^ a_|_) v a)))
4		an12 74	. . 3 ((a v b_|_) ^ (((b_|_ v a_|_) ^ (b v a_|_)) ^ ((b ^ a_|_) v a))) = (((b_|_ v a_|_) ^ (b v a_|_)) ^ ((a v b_|_) ^ ((b ^ a_|_) v a)))
5		ax-a2 30	. . . . 5 (b_|_ v a_|_) = (a_|_ v b_|_)
6		ax-a2 30	. . . . 5 (b v a_|_) = (a_|_ v b)
7	5, 6	2an 72	. . . 4 ((b_|_ v a_|_) ^ (b v a_|_)) = ((a_|_ v b_|_) ^ (a_|_ v b))
8		comor2 444	. . . . . . . . 9 (a v b_|_) C b_|_
9	8	comcom3 436	. . . . . . . 8 (a v b_|_)_|_ C b_|_
10	9	comcom5 440	. . . . . . 7 (a v b_|_) C b
11		comor1 443	. . . . . . . 8 (a v b_|_) C a
12	11	comcom2 175	. . . . . . 7 (a v b_|_) C a_|_
13	10, 12	com2an 466	. . . . . 6 (a v b_|_) C (b ^ a_|_)
14	13, 11	fh1 451	. . . . 5 ((a v b_|_) ^ ((b ^ a_|_) v a)) = (((a v b_|_) ^ (b ^ a_|_)) v ((a v b_|_) ^ a))
15		ax-a2 30	. . . . . . . . 9 (a v b_|_) = (b_|_ v a)
16		anor1 80	. . . . . . . . 9 (b ^ a_|_) = (b_|_ v a)_|_
17	15, 16	2an 72	. . . . . . . 8 ((a v b_|_) ^ (b ^ a_|_)) = ((b_|_ v a) ^ (b_|_ v a)_|_)
18		dff 93	. . . . . . . . 9 0 = ((b_|_ v a) ^ (b_|_ v a)_|_)
19	18	ax-r1 34	. . . . . . . 8 ((b_|_ v a) ^ (b_|_ v a)_|_) = 0
20	17, 19	ax-r2 35	. . . . . . 7 ((a v b_|_) ^ (b ^ a_|_)) = 0
21		ancom 68	. . . . . . . 8 ((a v b_|_) ^ a) = (a ^ (a v b_|_))
22		a5c 113	. . . . . . . 8 (a ^ (a v b_|_)) = a
23	21, 22	ax-r2 35	. . . . . . 7 ((a v b_|_) ^ a) = a
24	20, 23	2or 67	. . . . . 6 (((a v b_|_) ^ (b ^ a_|_)) v ((a v b_|_) ^ a)) = (0 v a)
25		ax-a2 30	. . . . . . 7 (0 v a) = (a v 0)
26		or0 94	. . . . . . 7 (a v 0) = a
27	25, 26	ax-r2 35	. . . . . 6 (0 v a) = a
28	24, 27	ax-r2 35	. . . . 5 (((a v b_|_) ^ (b ^ a_|_)) v ((a v b_|_) ^ a)) = a
29	14, 28	ax-r2 35	. . . 4 ((a v b_|_) ^ ((b ^ a_|_) v a)) = a
30	7, 29	2an 72	. . 3 (((b_|_ v a_|_) ^ (b v a_|_)) ^ ((a v b_|_) ^ ((b ^ a_|_) v a))) = (((a_|_ v b_|_) ^ (a_|_ v b)) ^ a)
31	4, 30	ax-r2 35	. 2 ((a v b_|_) ^ (((b_|_ v a_|_) ^ (b v a_|_)) ^ ((b ^ a_|_) v a))) = (((a_|_ v b_|_) ^ (a_|_ v b)) ^ a)
32	3, 31	ax-r2 35	1 (((a ->4 b)_|_ v (b ->4 a)) ^ (b ->4 a)_|_) = (((a_|_ v b_|_) ^ (a_|_ v b)) ^ a)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->4 wi4 16

This theorem is referenced by:  ud4lem1 563

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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