Theorem oi3ai3 485

Description: Theorem for Kalmbach implication.

Assertion

Ref	Expression
oi3ai3	((a ^ b) v (a ->3 b)_|_) = ((a v b) ^ (a_|_ ->3 b_|_))

Proof of Theorem oi3ai3

Step	Hyp	Ref	Expression
1		lea 152	. . . . . 6 (a ^ b) =< a
2		leo 150	. . . . . 6 a =< (a v b)
3	1, 2	letr 129	. . . . 5 (a ^ b) =< (a v b)
4	3	lecom 172	. . . 4 (a ^ b) C (a v b)
5		coman1 177	. . . . . 6 (a ^ b) C a
6		ancom 68	. . . . . . . 8 (a ^ b) = (b ^ a)
7		coman1 177	. . . . . . . 8 (b ^ a) C b
8	6, 7	bctr 173	. . . . . . 7 (a ^ b) C b
9	8	comcom2 175	. . . . . 6 (a ^ b) C b_|_
10	5, 9	com2an 466	. . . . 5 (a ^ b) C (a ^ b_|_)
11	5	comcom2 175	. . . . . 6 (a ^ b) C a_|_
12	5, 9	com2or 465	. . . . . 6 (a ^ b) C (a v b_|_)
13	11, 12	com2an 466	. . . . 5 (a ^ b) C (a_|_ ^ (a v b_|_))
14	10, 13	com2or 465	. . . 4 (a ^ b) C ((a ^ b_|_) v (a_|_ ^ (a v b_|_)))
15	4, 14	fh3 453	. . 3 ((a ^ b) v ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))) = (((a ^ b) v (a v b)) ^ ((a ^ b) v ((a ^ b_|_) v (a_|_ ^ (a v b_|_)))))
16	3	df-le2 123	. . . 4 ((a ^ b) v (a v b)) = (a v b)
17		ax-a3 31	. . . . . 6 (((a ^ b) v (a ^ b_|_)) v (a_|_ ^ (a v b_|_))) = ((a ^ b) v ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))
18	17	ax-r1 34	. . . . 5 ((a ^ b) v ((a ^ b_|_) v (a_|_ ^ (a v b_|_)))) = (((a ^ b) v (a ^ b_|_)) v (a_|_ ^ (a v b_|_)))
19		ax-a2 30	. . . . . 6 ((a ^ b) v (a ^ b_|_)) = ((a ^ b_|_) v (a ^ b))
20	19	ax-r5 37	. . . . 5 (((a ^ b) v (a ^ b_|_)) v (a_|_ ^ (a v b_|_))) = (((a ^ b_|_) v (a ^ b)) v (a_|_ ^ (a v b_|_)))
21	18, 20	ax-r2 35	. . . 4 ((a ^ b) v ((a ^ b_|_) v (a_|_ ^ (a v b_|_)))) = (((a ^ b_|_) v (a ^ b)) v (a_|_ ^ (a v b_|_)))
22	16, 21	2an 72	. . 3 (((a ^ b) v (a v b)) ^ ((a ^ b) v ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))) = ((a v b) ^ (((a ^ b_|_) v (a ^ b)) v (a_|_ ^ (a v b_|_))))
23	15, 22	ax-r2 35	. 2 ((a ^ b) v ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))) = ((a v b) ^ (((a ^ b_|_) v (a ^ b)) v (a_|_ ^ (a v b_|_))))
24		ni32 484	. . 3 (a ->3 b)_|_ = ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))
25	24	lor 66	. 2 ((a ^ b) v (a ->3 b)_|_) = ((a ^ b) v ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_)))))
26		i3n1 241	. . 3 (a_|_ ->3 b_|_) = (((a ^ b_|_) v (a ^ b)) v (a_|_ ^ (a v b_|_)))
27	26	lan 70	. 2 ((a v b) ^ (a_|_ ->3 b_|_)) = ((a v b) ^ (((a ^ b_|_) v (a ^ b)) v (a_|_ ^ (a v b_|_))))
28	23, 25, 27	3tr1 60	1 ((a ^ b) v (a ->3 b)_|_) = ((a v b) ^ (a_|_ ->3 b_|_))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->3 wi3 15

This theorem is referenced by:  i3orlem6 539

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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