undif3VD - Mathbox for Alan Sare

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Theorem undif3VD 39118

Description: The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 3888. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. undif3 3888 is undif3VD 39118 without virtual deductions and was automatically derived from undif3VD 39118.

1::	$\$ $\/$ $\$
2::	$\$ $/\$
3:2:	$\/$ $\$ $\/$ $/\$
4:1,3:	$\$ $\/$ $/\$
5::
6:5:	$\/$
7:5:	$\/$
8:6,7:	$\/$ $/\$ $\/$
9:8:	$\/$ $/\$ $\/$
10::	$/\$ $/\$
11:10:	$/\$
12:10:	$/\$
13:11:	$/\$ $\/$
14:12:	$/\$ $\/$
15:13,14:	$/\$ $\/$ $/\$ $\/$
16:15:	$/\$ $\/$ $/\$ $\/$
17:9,16:	$\/$ $/\$ $\/$ $/\$ $\/$
18::	$/\$ $/\$
19:18:	$/\$
20:18:	$/\$
21:18:	$/\$ $\/$ $/\$
22:21:	$/\$ $\/$ $/\$
23::	$/\$ $/\$
24:23:	$/\$
25:24:	$/\$ $\/$ $/\$
26:25:	$/\$ $\/$ $/\$
27:10:	$/\$ $\/$ $/\$
28:27:	$/\$ $\/$ $/\$
29::	$/\$ $/\$
30:29:	$/\$
31:30:	$/\$ $\/$ $/\$
32:31:	$/\$ $\/$ $/\$
33:22,26:	$/\$ $\/$ $/\$ $\/$ $/\$
34:28,32:	$/\$ $\/$ $/\$ $\/$ $/\$
35:33,34:	$/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36::	$/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$
37:36,35:	$\/$ $/\$ $\/$ $\/$ $/\$
38:17,37:	$\/$ $/\$ $\/$ $/\$ $\/$
39::	$\$ $/\$
40:39:	$\$ $/\$
41::	$/\$ $\/$
42:40,41:	$\$ $\/$
43::	$\/$
44:43,42:	$/\$ $\$ $\/$ $/\$ $/\$
45::	$\$ $\$ $/\$ $\$
46:45,44:	$\$ $\$ $\/$ $/\$ $\/$
47:4,38:	$\$ $\/$ $/\$ $\/$
48:46,47:	$\$ $\$ $\$
49:48:	$\$ $\$ $\$
qed:49:	$\$ $\$ $\$

(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

**Assertion**
Ref	Expression
undif3VD	$\$ $\$ $\$

Proof of Theorem undif3VD Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 3753	. . . . . 6 $\$ $\/$ $\$
2		eldif 3584	. . . . . . 7 $\$ $/\$
3	2	orbi2i 541	. . . . . 6 $\/$ $\$ $\/$ $/\$
4	1, 3	bitri 264	. . . . 5 $\$ $\/$ $/\$
5		idn1 38790	. . . . . . . . . 10
6		orc 400	. . . . . . . . . 10 $\/$
7	5, 6	e1a 38852	. . . . . . . . 9 $\/$
8		olc 399	. . . . . . . . . 10 $\/$
9	5, 8	e1a 38852	. . . . . . . . 9 $\/$
10		pm3.2 463	. . . . . . . . 9 $\/$ $\/$ $\/$ $/\$ $\/$
11	7, 9, 10	e11 38913	. . . . . . . 8 $\/$ $/\$ $\/$
12	11	in1 38787	. . . . . . 7 $\/$ $/\$ $\/$
13		idn1 38790	. . . . . . . . . . 11 $/\$ $/\$
14		simpl 473	. . . . . . . . . . 11 $/\$
15	13, 14	e1a 38852	. . . . . . . . . 10 $/\$
16		olc 399	. . . . . . . . . 10 $\/$
17	15, 16	e1a 38852	. . . . . . . . 9 $/\$ $\/$
18		simpr 477	. . . . . . . . . . 11 $/\$
19	13, 18	e1a 38852	. . . . . . . . . 10 $/\$
20		orc 400	. . . . . . . . . 10 $\/$
21	19, 20	e1a 38852	. . . . . . . . 9 $/\$ $\/$
22	17, 21, 10	e11 38913	. . . . . . . 8 $/\$ $\/$ $/\$ $\/$
23	22	in1 38787	. . . . . . 7 $/\$ $\/$ $/\$ $\/$
24	12, 23	jaoi 394	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$
25		anddi 914	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	25	bicomi 214	. . . . . . 7 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$
27		idn1 38790	. . . . . . . . . . 11 $/\$ $/\$
28		simpl 473	. . . . . . . . . . . 12 $/\$
29	28	orcd 407	. . . . . . . . . . 11 $/\$ $\/$ $/\$
30	27, 29	e1a 38852	. . . . . . . . . 10 $/\$ $\/$ $/\$
31	30	in1 38787	. . . . . . . . 9 $/\$ $\/$ $/\$
32		idn1 38790	. . . . . . . . . . . 12 $/\$ $/\$
33		simpl 473	. . . . . . . . . . . 12 $/\$
34	32, 33	e1a 38852	. . . . . . . . . . 11 $/\$
35		orc 400	. . . . . . . . . . 11 $\/$ $/\$
36	34, 35	e1a 38852	. . . . . . . . . 10 $/\$ $\/$ $/\$
37	36	in1 38787	. . . . . . . . 9 $/\$ $\/$ $/\$
38	31, 37	jaoi 394	. . . . . . . 8 $/\$ $\/$ $/\$ $\/$ $/\$
39		olc 399	. . . . . . . . . . 11 $/\$ $\/$ $/\$
40	13, 39	e1a 38852	. . . . . . . . . 10 $/\$ $\/$ $/\$
41	40	in1 38787	. . . . . . . . 9 $/\$ $\/$ $/\$
42		idn1 38790	. . . . . . . . . . . 12 $/\$ $/\$
43		simpr 477	. . . . . . . . . . . 12 $/\$
44	42, 43	e1a 38852	. . . . . . . . . . 11 $/\$
45	44, 35	e1a 38852	. . . . . . . . . 10 $/\$ $\/$ $/\$
46	45	in1 38787	. . . . . . . . 9 $/\$ $\/$ $/\$
47	41, 46	jaoi 394	. . . . . . . 8 $/\$ $\/$ $/\$ $\/$ $/\$
48	38, 47	jaoi 394	. . . . . . 7 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
49	26, 48	sylbir 225	. . . . . 6 $\/$ $/\$ $\/$ $\/$ $/\$
50	24, 49	impbii 199	. . . . 5 $\/$ $/\$ $\/$ $/\$ $\/$
51	4, 50	bitri 264	. . . 4 $\$ $\/$ $/\$ $\/$
52		eldif 3584	. . . . 5 $\$ $\$ $/\$ $\$
53		elun 3753	. . . . . 6 $\/$
54		eldif 3584	. . . . . . . 8 $\$ $/\$
55	54	notbii 310	. . . . . . 7 $\$ $/\$
56		pm4.53 513	. . . . . . 7 $/\$ $\/$
57	55, 56	bitri 264	. . . . . 6 $\$ $\/$
58	53, 57	anbi12i 733	. . . . 5 $/\$ $\$ $\/$ $/\$ $\/$
59	52, 58	bitri 264	. . . 4 $\$ $\$ $\/$ $/\$ $\/$
60	51, 59	bitr4i 267	. . 3 $\$ $\$ $\$
61	60	ax-gen 1722	. 2 $\$ $\$ $\$
62		dfcleq 2616	. . 3 $\$ $\$ $\$ $\$ $\$ $\$
63	62	biimpri 218	. 2 $\$ $\$ $\$ $\$ $\$ $\$
64	61, 63	e0a 38999	1 $\$ $\$ $\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wb 196 $\/$ wo 383 $/\$ wa 384

wal 1481

wceq 1483

wcel 1990 $\$ cdif 3571

cun 3572

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-vd1 38786

This theorem is referenced by: (None)

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