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Theorem 2pm13.193VD 33571
Description: Virtual deduction proof of 2pm13.193 33193. 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 33193 is 2pm13.193VD 33571 without virtual deductions and was automatically derived from 2pm13.193VD 33571. (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 33219 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) ).
2 simpl 457 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( x  =  u  /\  y  =  v ) )
31, 2e1a 33281 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( x  =  u  /\  y  =  v ) ).
4 simpl 457 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  v )  ->  x  =  u )
53, 4e1a 33281 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  x  =  u ).
6 simpr 461 . . . . . . . . . . 11  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  [ u  /  x ] [ v  / 
y ] ph )
71, 6e1a 33281 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  [ u  /  x ] [ v  /  y ] ph ).
8 pm3.21 448 . . . . . . . . . 10  |-  ( x  =  u  ->  ( [ u  /  x ] [ v  /  y ] ph  ->  ( [
u  /  x ] [ v  /  y ] ph  /\  x  =  u ) ) )
95, 7, 8e11 33342 . . . . . . . . 9  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ u  /  x ] [ v  / 
y ] ph  /\  x  =  u ) ).
10 sbequ2 1728 . . . . . . . . . 10  |-  ( x  =  u  ->  ( [ u  /  x ] [ v  /  y ] ph  ->  [ v  /  y ] ph ) )
1110imdistanri 691 . . . . . . . . 9  |-  ( ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u )  ->  ( [ v  /  y ] ph  /\  x  =  u ) )
129, 11e1a 33281 . . . . . . . 8  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ v  / 
y ] ph  /\  x  =  u ) ).
13 simpl 457 . . . . . . . 8  |-  ( ( [ v  /  y ] ph  /\  x  =  u )  ->  [ v  /  y ] ph )
1412, 13e1a 33281 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  [ v  /  y ] ph ).
15 simpr 461 . . . . . . . 8  |-  ( ( x  =  u  /\  y  =  v )  ->  y  =  v )
163, 15e1a 33281 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  y  =  v ).
17 pm3.2 447 . . . . . . 7  |-  ( [ v  /  y ]
ph  ->  ( y  =  v  ->  ( [
v  /  y ]
ph  /\  y  =  v ) ) )
1814, 16, 17e11 33342 . . . . . 6  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ v  / 
y ] ph  /\  y  =  v ) ).
19 sbequ2 1728 . . . . . . 7  |-  ( y  =  v  ->  ( [ v  /  y ] ph  ->  ph ) )
2019imdistanri 691 . . . . . 6  |-  ( ( [ v  /  y ] ph  /\  y  =  v )  ->  ( ph  /\  y  =  v ) )
2118, 20e1a 33281 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ph  /\  y  =  v ) ).
22 simpl 457 . . . . 5  |-  ( (
ph  /\  y  =  v )  ->  ph )
2321, 22e1a 33281 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ph ).
24 pm3.2 447 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  ->  ( ph  ->  (
( x  =  u  /\  y  =  v )  /\  ph )
) )
253, 23, 24e11 33342 . . 3  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ( x  =  u  /\  y  =  v )  /\  ph ) ).
2625in1 33216 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
27 idn1 33219 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( ( x  =  u  /\  y  =  v )  /\  ph ) ).
28 simpl 457 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ( x  =  u  /\  y  =  v ) )
2927, 28e1a 33281 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( x  =  u  /\  y  =  v ) ).
3029, 4e1a 33281 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  x  =  u ).
3129, 15e1a 33281 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  y  =  v ).
32 simpr 461 . . . . . . . . . . 11  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ph )
3327, 32e1a 33281 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ph ).
34 pm3.21 448 . . . . . . . . . 10  |-  ( y  =  v  ->  ( ph  ->  ( ph  /\  y  =  v )
) )
3531, 33, 34e11 33342 . . . . . . . . 9  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  (
ph  /\  y  =  v ) ).
36 sbequ1 1977 . . . . . . . . . 10  |-  ( y  =  v  ->  ( ph  ->  [ v  / 
y ] ph )
)
3736imdistanri 691 . . . . . . . . 9  |-  ( (
ph  /\  y  =  v )  ->  ( [ v  /  y ] ph  /\  y  =  v ) )
3835, 37e1a 33281 . . . . . . . 8  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( [ v  /  y ] ph  /\  y  =  v ) ).
39 simpl 457 . . . . . . . 8  |-  ( ( [ v  /  y ] ph  /\  y  =  v )  ->  [ v  /  y ] ph )
4038, 39e1a 33281 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  [ v  /  y ]
ph ).
41 pm3.21 448 . . . . . . 7  |-  ( x  =  u  ->  ( [ v  /  y ] ph  ->  ( [
v  /  y ]
ph  /\  x  =  u ) ) )
4230, 40, 41e11 33342 . . . . . 6  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( [ v  /  y ] ph  /\  x  =  u ) ).
43 sbequ1 1977 . . . . . . 7  |-  ( x  =  u  ->  ( [ v  /  y ] ph  ->  [ u  /  x ] [ v  /  y ] ph ) )
4443imdistanri 691 . . . . . 6  |-  ( ( [ v  /  y ] ph  /\  x  =  u )  ->  ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u ) )
4542, 44e1a 33281 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u ) ).
46 simpl 457 . . . . 5  |-  ( ( [ u  /  x ] [ v  /  y ] ph  /\  x  =  u )  ->  [ u  /  x ] [ v  /  y ] ph )
4745, 46e1a 33281 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  [ u  /  x ] [ v  /  y ] ph ).
48 pm3.2 447 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  ->  ( [ u  /  x ] [ v  / 
y ] ph  ->  ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) ) )
4929, 47, 48e11 33342 . . 3  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) ).
5049in1 33216 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) )
5126, 50impbii 188 1  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1383   [wsb 1726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-12 1840
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1600  df-sb 1727  df-vd1 33215
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator