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Theorem 2pm13.193VD 28349
Description: Virtual deduction proof of 2pm13.193 27975. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. 2pm13.193 27975 is 2pm13.193VD 28349 without virtual deductions and was automatically derived from 2pm13.193VD 28349. (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 27999 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) ).
2 simpl 444 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( x  =  u  /\  y  =  v ) )
31, 2e1_ 28062 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( x  =  u  /\  y  =  v ) ).
4 simpl 444 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  v )  ->  x  =  u )
53, 4e1_ 28062 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  x  =  u ).
6 simpr 448 . . . . . . . . . . 11  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  [ u  /  x ] [ v  / 
y ] ph )
71, 6e1_ 28062 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  [ u  /  x ] [ v  /  y ] ph ).
8 pm3.21 436 . . . . . . . . . 10  |-  ( x  =  u  ->  ( [ u  /  x ] [ v  /  y ] ph  ->  ( [
u  /  x ] [ v  /  y ] ph  /\  x  =  u ) ) )
95, 7, 8e11 28123 . . . . . . . . 9  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ u  /  x ] [ v  / 
y ] ph  /\  x  =  u ) ).
10 sbequ2 1657 . . . . . . . . . 10  |-  ( x  =  u  ->  ( [ u  /  x ] [ v  /  y ] ph  ->  [ v  /  y ] ph ) )
1110imdistanri 673 . . . . . . . . 9  |-  ( ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u )  ->  ( [ v  /  y ] ph  /\  x  =  u ) )
129, 11e1_ 28062 . . . . . . . 8  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ v  / 
y ] ph  /\  x  =  u ) ).
13 simpl 444 . . . . . . . 8  |-  ( ( [ v  /  y ] ph  /\  x  =  u )  ->  [ v  /  y ] ph )
1412, 13e1_ 28062 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  [ v  /  y ] ph ).
15 simpr 448 . . . . . . . 8  |-  ( ( x  =  u  /\  y  =  v )  ->  y  =  v )
163, 15e1_ 28062 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  y  =  v ).
17 pm3.2 435 . . . . . . 7  |-  ( [ v  /  y ]
ph  ->  ( y  =  v  ->  ( [
v  /  y ]
ph  /\  y  =  v ) ) )
1814, 16, 17e11 28123 . . . . . 6  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ v  / 
y ] ph  /\  y  =  v ) ).
19 sbequ2 1657 . . . . . . 7  |-  ( y  =  v  ->  ( [ v  /  y ] ph  ->  ph ) )
2019imdistanri 673 . . . . . 6  |-  ( ( [ v  /  y ] ph  /\  y  =  v )  ->  ( ph  /\  y  =  v ) )
2118, 20e1_ 28062 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ph  /\  y  =  v ) ).
22 simpl 444 . . . . 5  |-  ( (
ph  /\  y  =  v )  ->  ph )
2321, 22e1_ 28062 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ph ).
24 pm3.2 435 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  ->  ( ph  ->  (
( x  =  u  /\  y  =  v )  /\  ph )
) )
253, 23, 24e11 28123 . . 3  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ( x  =  u  /\  y  =  v )  /\  ph ) ).
2625in1 27996 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
27 idn1 27999 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( ( x  =  u  /\  y  =  v )  /\  ph ) ).
28 simpl 444 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ( x  =  u  /\  y  =  v ) )
2927, 28e1_ 28062 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( x  =  u  /\  y  =  v ) ).
3029, 4e1_ 28062 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  x  =  u ).
3129, 15e1_ 28062 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  y  =  v ).
32 simpr 448 . . . . . . . . . . 11  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ph )
3327, 32e1_ 28062 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ph ).
34 pm3.21 436 . . . . . . . . . 10  |-  ( y  =  v  ->  ( ph  ->  ( ph  /\  y  =  v )
) )
3531, 33, 34e11 28123 . . . . . . . . 9  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  (
ph  /\  y  =  v ) ).
36 sbequ1 1932 . . . . . . . . . 10  |-  ( y  =  v  ->  ( ph  ->  [ v  / 
y ] ph )
)
3736imdistanri 673 . . . . . . . . 9  |-  ( (
ph  /\  y  =  v )  ->  ( [ v  /  y ] ph  /\  y  =  v ) )
3835, 37e1_ 28062 . . . . . . . 8  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( [ v  /  y ] ph  /\  y  =  v ) ).
39 simpl 444 . . . . . . . 8  |-  ( ( [ v  /  y ] ph  /\  y  =  v )  ->  [ v  /  y ] ph )
4038, 39e1_ 28062 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  [ v  /  y ]
ph ).
41 pm3.21 436 . . . . . . 7  |-  ( x  =  u  ->  ( [ v  /  y ] ph  ->  ( [
v  /  y ]
ph  /\  x  =  u ) ) )
4230, 40, 41e11 28123 . . . . . 6  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( [ v  /  y ] ph  /\  x  =  u ) ).
43 sbequ1 1932 . . . . . . 7  |-  ( x  =  u  ->  ( [ v  /  y ] ph  ->  [ u  /  x ] [ v  /  y ] ph ) )
4443imdistanri 673 . . . . . 6  |-  ( ( [ v  /  y ] ph  /\  x  =  u )  ->  ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u ) )
4542, 44e1_ 28062 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u ) ).
46 simpl 444 . . . . 5  |-  ( ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u )  ->  [ u  /  x ] [ v  /  y ] ph )
4745, 46e1_ 28062 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  [ u  /  x ] [ v  /  y ] ph ).
48 pm3.2 435 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  ->  ( [ u  /  x ] [ v  / 
y ] ph  ->  ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) ) )
4929, 47, 48e11 28123 . . 3  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) ).
5049in1 27996 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) )
5126, 50impbii 181 1  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649   [wsb 1655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-sb 1656  df-vd1 27995
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