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Theorem 2pm13.193VD 28767
Description: Virtual deduction proof of 2pm13.193 28392. 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 28392 is 2pm13.193VD 28767 without virtual deductions and was automatically derived from 2pm13.193VD 28767. (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 28417 . . . . 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_ 28480 . . . 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_ 28480 . . . . . . . . . 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_ 28480 . . . . . . . . . 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 28541 . . . . . . . . 9  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ u  /  x ] [ v  / 
y ] ph  /\  x  =  u ) ).
10 sbequ2 1660 . . . . . . . . . 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_ 28480 . . . . . . . 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_ 28480 . . . . . . 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_ 28480 . . . . . . 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 28541 . . . . . 6  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( [ v  / 
y ] ph  /\  y  =  v ) ).
19 sbequ2 1660 . . . . . . 7  |-  ( y  =  v  ->  ( [ v  /  y ] ph  ->  ph ) )
2019imdistanri 673 . . . . . 6  |-  ( ( [ v  /  y ] ph  /\  y  =  v )  ->  ( ph  /\  y  =  v ) )
2118, 20e1_ 28480 . . . . 5  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ph  /\  y  =  v ) ).
22 simpl 444 . . . . 5  |-  ( (
ph  /\  y  =  v )  ->  ph )
2321, 22e1_ 28480 . . . 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 28541 . . 3  |-  (. (
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
->.  ( ( x  =  u  /\  y  =  v )  /\  ph ) ).
2625in1 28414 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
27 idn1 28417 . . . . 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_ 28480 . . . 4  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( x  =  u  /\  y  =  v ) ).
3029, 4e1_ 28480 . . . . . . 7  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  x  =  u ).
3129, 15e1_ 28480 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  y  =  v ).
32 simpr 448 . . . . . . . . . . 11  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ph )
3327, 32e1_ 28480 . . . . . . . . . 10  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ph ).
34 pm3.21 436 . . . . . . . . . 10  |-  ( y  =  v  ->  ( ph  ->  ( ph  /\  y  =  v )
) )
3531, 33, 34e11 28541 . . . . . . . . 9  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  (
ph  /\  y  =  v ) ).
36 sbequ1 1943 . . . . . . . . . 10  |-  ( y  =  v  ->  ( ph  ->  [ v  / 
y ] ph )
)
3736imdistanri 673 . . . . . . . . 9  |-  ( (
ph  /\  y  =  v )  ->  ( [ v  /  y ] ph  /\  y  =  v ) )
3835, 37e1_ 28480 . . . . . . . 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_ 28480 . . . . . . 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 28541 . . . . . 6  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( [ v  /  y ] ph  /\  x  =  u ) ).
43 sbequ1 1943 . . . . . . 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_ 28480 . . . . 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_ 28480 . . . 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 28541 . . 3  |-  (. (
( x  =  u  /\  y  =  v )  /\  ph )  ->.  ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) ).
5049in1 28414 . 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 1652   [wsb 1658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-sb 1659  df-vd1 28413
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