Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2pm13.193VD Structured version   Visualization version   Unicode version

Theorem 2pm13.193VD 37300
Description: Virtual deduction proof of 2pm13.193 36919. 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 36919 is 2pm13.193VD 37300 without virtual deductions and was automatically derived from 2pm13.193VD 37300. (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
StepHypRef Expression
1 idn1 36944 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) ).
2 simpl 459 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( x  =  u  /\  y  =  v ) )
31, 2e1a 37006 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( x  =  u  /\  y  =  v ) ).
4 simpl 459 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  v )  ->  x  =  u )
53, 4e1a 37006 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  x  =  u ).
6 simpr 463 . . . . . . . . . . 11  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  [ u  /  x ] [ v  / 
y ] ph )
71, 6e1a 37006 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  [ u  /  x ] [ v  /  y ] ph ).
8 pm3.21 450 . . . . . . . . . 10  |-  ( x  =  u  ->  ( [ u  /  x ] [ v  /  y ] ph  ->  ( [
u  /  x ] [ v  /  y ] ph  /\  x  =  u ) ) )
95, 7, 8e11 37067 . . . . . . . . 9  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ u  /  x ] [ v  / 
y ] ph  /\  x  =  u ) ).
10 sbequ2 1799 . . . . . . . . . 10  |-  ( x  =  u  ->  ( [ u  /  x ] [ v  /  y ] ph  ->  [ v  /  y ] ph ) )
1110imdistanri 697 . . . . . . . . 9  |-  ( ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u )  ->  ( [ v  /  y ] ph  /\  x  =  u ) )
129, 11e1a 37006 . . . . . . . 8  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ v  / 
y ] ph  /\  x  =  u ) ).
13 simpl 459 . . . . . . . 8  |-  ( ( [ v  /  y ] ph  /\  x  =  u )  ->  [ v  /  y ] ph )
1412, 13e1a 37006 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  [ v  /  y ] ph ).
15 simpr 463 . . . . . . . 8  |-  ( ( x  =  u  /\  y  =  v )  ->  y  =  v )
163, 15e1a 37006 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  y  =  v ).
17 pm3.2 449 . . . . . . 7  |-  ( [ v  /  y ]
ph  ->  ( y  =  v  ->  ( [
v  /  y ]
ph  /\  y  =  v ) ) )
1814, 16, 17e11 37067 . . . . . 6  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ v  / 
y ] ph  /\  y  =  v ) ).
19 sbequ2 1799 . . . . . . 7  |-  ( y  =  v  ->  ( [ v  /  y ] ph  ->  ph ) )
2019imdistanri 697 . . . . . 6  |-  ( ( [ v  /  y ] ph  /\  y  =  v )  ->  ( ph  /\  y  =  v ) )
2118, 20e1a 37006 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ph  /\  y  =  v ) ).
22 simpl 459 . . . . 5  |-  ( (
ph  /\  y  =  v )  ->  ph )
2321, 22e1a 37006 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ph ).
24 pm3.2 449 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  ->  ( ph  ->  (
( x  =  u  /\  y  =  v )  /\  ph )
) )
253, 23, 24e11 37067 . . 3  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ( x  =  u  /\  y  =  v )  /\  ph ) ).
2625in1 36941 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
27 idn1 36944 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( ( x  =  u  /\  y  =  v )  /\  ph ) ).
28 simpl 459 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ( x  =  u  /\  y  =  v ) )
2927, 28e1a 37006 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( x  =  u  /\  y  =  v ) ).
3029, 4e1a 37006 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  x  =  u ).
3129, 15e1a 37006 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  y  =  v ).
32 simpr 463 . . . . . . . . . . 11  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ph )
3327, 32e1a 37006 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ph ).
34 pm3.21 450 . . . . . . . . . 10  |-  ( y  =  v  ->  ( ph  ->  ( ph  /\  y  =  v )
) )
3531, 33, 34e11 37067 . . . . . . . . 9  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  (
ph  /\  y  =  v ) ).
36 sbequ1 2082 . . . . . . . . . 10  |-  ( y  =  v  ->  ( ph  ->  [ v  / 
y ] ph )
)
3736imdistanri 697 . . . . . . . . 9  |-  ( (
ph  /\  y  =  v )  ->  ( [ v  /  y ] ph  /\  y  =  v ) )
3835, 37e1a 37006 . . . . . . . 8  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( [ v  /  y ] ph  /\  y  =  v ) ).
39 simpl 459 . . . . . . . 8  |-  ( ( [ v  /  y ] ph  /\  y  =  v )  ->  [ v  /  y ] ph )
4038, 39e1a 37006 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  [ v  /  y ]
ph ).
41 pm3.21 450 . . . . . . 7  |-  ( x  =  u  ->  ( [ v  /  y ] ph  ->  ( [
v  /  y ]
ph  /\  x  =  u ) ) )
4230, 40, 41e11 37067 . . . . . 6  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( [ v  /  y ] ph  /\  x  =  u ) ).
43 sbequ1 2082 . . . . . . 7  |-  ( x  =  u  ->  ( [ v  /  y ] ph  ->  [ u  /  x ] [ v  /  y ] ph ) )
4443imdistanri 697 . . . . . 6  |-  ( ( [ v  /  y ] ph  /\  x  =  u )  ->  ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u ) )
4542, 44e1a 37006 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u ) ).
46 simpl 459 . . . . 5  |-  ( ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u )  ->  [ u  /  x ] [ v  /  y ] ph )
4745, 46e1a 37006 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  [ u  /  x ] [ v  /  y ] ph ).
48 pm3.2 449 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  ->  ( [ u  /  x ] [ v  / 
y ] ph  ->  ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) ) )
4929, 47, 48e11 37067 . . 3  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) ).
5049in1 36941 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) )
5126, 50impbii 191 1  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1444   [wsb 1797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-12 1933
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-sb 1798  df-vd1 36940
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator