Theorem 2pm13.193VD 39139

Description: Virtual deduction proof of 2pm13.193 38768. 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. 2pm13.193 38768 is 2pm13.193VD 39139 without virtual deductions and was automatically derived from 2pm13.193VD 39139. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

1::	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ).
2:1:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( x = u /\ y = v ) ).
3:2:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. x = u ).
4:1:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. [ u / x ] [ v / y ] ph ).
5:3,4:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( [ u / x ] [ v / y ] ph /\ x = u ) ).
6:5:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( [ v / y ] ph /\ x = u ) ).
7:6:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. [ v / y ] ph ).
8:2:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. y = v ).
9:7,8:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( [ v / y ] ph /\ y = v ) ).
10:9:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( ph /\ y = v ) ).
11:10:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ph ).
12:2,11:	|- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( ( x = u /\ y = v ) /\ ph ) ).
13:12:	|- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) -> ( ( x = u /\ y = v ) /\ ph ) )
14::	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( ( x = u /\ y = v ) /\ ph ) ).
15:14:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( x = u /\ y = v ) ).
16:15:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. y = v ).
17:14:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. ph ).
18:16,17:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( ph /\ y = v ) ).
19:18:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( [ v / y ] ph /\ y = v ) ).
20:15:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. x = u ).
21:19:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. [ v / y ] ph ).
22:20,21:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( [ v / y ] ph /\ x = u ) ).
23:22:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( [ u / x ] [ v / y ] ph /\ x = u ) ).
24:23:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. [ u / x ] [ v / y ] ph ).
25:15,24:	|- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ).
26:25:	|- ( ( ( x = u /\ y = v ) /\ ph ) -> ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
qed:13,26:	|- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( ( x = u /\ y = v ) /\ ph ) )

Assertion

Ref	Expression
2pm13.193VD	|- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( ( x = u /\ y = v ) /\ ph ) )

Proof of Theorem 2pm13.193VD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . 5 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ).
2		simpl 473	. . . . 5 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) -> ( x = u /\ y = v ) )
3	1, 2	e1a 38852	. . . 4 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( x = u /\ y = v ) ).
4		simpl 473	. . . . . . . . . . 11 |- ( ( x = u /\ y = v ) -> x = u )
5	3, 4	e1a 38852	. . . . . . . . . 10 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. x = u ).
6		simpr 477	. . . . . . . . . . 11 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) -> [ u / x ] [ v / y ] ph )
7	1, 6	e1a 38852	. . . . . . . . . 10 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. [ u / x ] [ v / y ] ph ).
8		pm3.21 464	. . . . . . . . . 10 |- ( x = u -> ( [ u / x ] [ v / y ] ph -> ( [ u / x ] [ v / y ] ph /\ x = u ) ) )
9	5, 7, 8	e11 38913	. . . . . . . . 9 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( [ u / x ] [ v / y ] ph /\ x = u ) ).
10		sbequ2 1882	. . . . . . . . . 10 |- ( x = u -> ( [ u / x ] [ v / y ] ph -> [ v / y ] ph ) )
11	10	imdistanri 727	. . . . . . . . 9 |- ( ( [ u / x ] [ v / y ] ph /\ x = u ) -> ( [ v / y ] ph /\ x = u ) )
12	9, 11	e1a 38852	. . . . . . . 8 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( [ v / y ] ph /\ x = u ) ).
13		simpl 473	. . . . . . . 8 |- ( ( [ v / y ] ph /\ x = u ) -> [ v / y ] ph )
14	12, 13	e1a 38852	. . . . . . 7 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. [ v / y ] ph ).
15		simpr 477	. . . . . . . 8 |- ( ( x = u /\ y = v ) -> y = v )
16	3, 15	e1a 38852	. . . . . . 7 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. y = v ).
17		pm3.2 463	. . . . . . 7 |- ( [ v / y ] ph -> ( y = v -> ( [ v / y ] ph /\ y = v ) ) )
18	14, 16, 17	e11 38913	. . . . . 6 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( [ v / y ] ph /\ y = v ) ).
19		sbequ2 1882	. . . . . . 7 |- ( y = v -> ( [ v / y ] ph -> ph ) )
20	19	imdistanri 727	. . . . . 6 |- ( ( [ v / y ] ph /\ y = v ) -> ( ph /\ y = v ) )
21	18, 20	e1a 38852	. . . . 5 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( ph /\ y = v ) ).
22		simpl 473	. . . . 5 |- ( ( ph /\ y = v ) -> ph )
23	21, 22	e1a 38852	. . . 4 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ph ).
24		pm3.2 463	. . . 4 |- ( ( x = u /\ y = v ) -> ( ph -> ( ( x = u /\ y = v ) /\ ph ) ) )
25	3, 23, 24	e11 38913	. . 3 |- (. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ->. ( ( x = u /\ y = v ) /\ ph ) ).
26	25	in1 38787	. 2 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) -> ( ( x = u /\ y = v ) /\ ph ) )
27		idn1 38790	. . . . 5 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( ( x = u /\ y = v ) /\ ph ) ).
28		simpl 473	. . . . 5 |- ( ( ( x = u /\ y = v ) /\ ph ) -> ( x = u /\ y = v ) )
29	27, 28	e1a 38852	. . . 4 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( x = u /\ y = v ) ).
30	29, 4	e1a 38852	. . . . . . 7 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. x = u ).
31	29, 15	e1a 38852	. . . . . . . . . 10 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. y = v ).
32		simpr 477	. . . . . . . . . . 11 |- ( ( ( x = u /\ y = v ) /\ ph ) -> ph )
33	27, 32	e1a 38852	. . . . . . . . . 10 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. ph ).
34		pm3.21 464	. . . . . . . . . 10 |- ( y = v -> ( ph -> ( ph /\ y = v ) ) )
35	31, 33, 34	e11 38913	. . . . . . . . 9 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( ph /\ y = v ) ).
36		sbequ1 2110	. . . . . . . . . 10 |- ( y = v -> ( ph -> [ v / y ] ph ) )
37	36	imdistanri 727	. . . . . . . . 9 |- ( ( ph /\ y = v ) -> ( [ v / y ] ph /\ y = v ) )
38	35, 37	e1a 38852	. . . . . . . 8 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( [ v / y ] ph /\ y = v ) ).
39		simpl 473	. . . . . . . 8 |- ( ( [ v / y ] ph /\ y = v ) -> [ v / y ] ph )
40	38, 39	e1a 38852	. . . . . . 7 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. [ v / y ] ph ).
41		pm3.21 464	. . . . . . 7 |- ( x = u -> ( [ v / y ] ph -> ( [ v / y ] ph /\ x = u ) ) )
42	30, 40, 41	e11 38913	. . . . . 6 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( [ v / y ] ph /\ x = u ) ).
43		sbequ1 2110	. . . . . . 7 |- ( x = u -> ( [ v / y ] ph -> [ u / x ] [ v / y ] ph ) )
44	43	imdistanri 727	. . . . . 6 |- ( ( [ v / y ] ph /\ x = u ) -> ( [ u / x ] [ v / y ] ph /\ x = u ) )
45	42, 44	e1a 38852	. . . . 5 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( [ u / x ] [ v / y ] ph /\ x = u ) ).
46		simpl 473	. . . . 5 |- ( ( [ u / x ] [ v / y ] ph /\ x = u ) -> [ u / x ] [ v / y ] ph )
47	45, 46	e1a 38852	. . . 4 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. [ u / x ] [ v / y ] ph ).
48		pm3.2 463	. . . 4 |- ( ( x = u /\ y = v ) -> ( [ u / x ] [ v / y ] ph -> ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ) )
49	29, 47, 48	e11 38913	. . 3 |- (. ( ( x = u /\ y = v ) /\ ph ) ->. ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ).
50	49	in1 38787	. 2 |- ( ( ( x = u /\ y = v ) /\ ph ) -> ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
51	26, 50	impbii 199	1 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( ( x = u /\ y = v ) /\ ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   [wsb 1880

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-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-vd1 38786

This theorem is referenced by: (None)