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Theorem a9e2eqVD 28361
Description: The following User's Proof is a Virtual Deduction proof (see: wvd1 28002) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. a9e2eq 27988 is a9e2eqVD 28361 without virtual deductions and was automatically derived from a9e2eqVD 28361. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  (. A. x x  =  y  ->.  A. x x  =  y ).
2::  |-  (. A. x x  =  y ,. x  =  u  ->.  x  =  u ).
3:1:  |-  (. A. x x  =  y  ->.  x  =  y ).
4:2,3:  |-  (. A. x x  =  y ,. x  =  u  ->.  y  =  u ).
5:2,4:  |-  (. A. x x  =  y ,. x  =  u  ->.  ( x  =  u  /\  y  =  u ) ).
6:5:  |-  (. A. x x  =  y  ->.  ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) ).
7:6:  |-  ( A. x x  =  y  ->  ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) )
8:7:  |-  ( A. x A. x x  =  y  ->  A. x ( x  =  u  ->  (  x  =  u  /\  y  =  u ) ) )
9::  |-  ( A. x x  =  y  <->  A. x A. x x  =  y )
10:8,9:  |-  ( A. x x  =  y  ->  A. x ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) )
11:1,10:  |-  (. A. x x  =  y  ->.  A. x ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) ).
12:11:  |-  (. A. x x  =  y  ->.  ( E. x x  =  u  ->  E. x  ( x  =  u  /\  y  =  u ) ) ).
13::  |-  E. x x  =  u
14:13,12:  |-  (. A. x x  =  y  ->.  E. x ( x  =  u  /\  y  =  u  ) ).
140:14:  |-  ( A. x x  =  y  ->  E. x ( x  =  u  /\  y  =  u )  )
141:140:  |-  ( A. x x  =  y  ->  A. x E. x ( x  =  u  /\  y  =  u ) )
15:1,141:  |-  (. A. x x  =  y  ->.  A. x E. x ( x  =  u  /\  y  =  u ) ).
16:1,15:  |-  (. A. x x  =  y  ->.  A. y E. x ( x  =  u  /\  y  =  u ) ).
17:16:  |-  (. A. x x  =  y  ->.  E. y E. x ( x  =  u  /\  y  =  u ) ).
18:17:  |-  (. A. x x  =  y  ->.  E. x E. y ( x  =  u  /\  y  =  u ) ).
19::  |-  (. u  =  v  ->.  u  =  v ).
20::  |-  (. u  =  v ,. ( x  =  u  /\  y  =  u )  ->.  ( x  =  u  /\  y  =  u ) ).
21:20:  |-  (. u  =  v ,. ( x  =  u  /\  y  =  u )  ->.  y  =  u  ).
22:19,21:  |-  (. u  =  v ,. ( x  =  u  /\  y  =  u )  ->.  y  =  v  ).
23:20:  |-  (. u  =  v ,. ( x  =  u  /\  y  =  u )  ->.  x  =  u  ).
24:22,23:  |-  (. u  =  v ,. ( x  =  u  /\  y  =  u )  ->.  ( x  =  u  /\  y  =  v ) ).
25:24:  |-  (. u  =  v  ->.  ( ( x  =  u  /\  y  =  u )  ->  (  x  =  u  /\  y  =  v ) ) ).
26:25:  |-  (. u  =  v  ->.  A. y ( ( x  =  u  /\  y  =  u )  ->  ( x  =  u  /\  y  =  v ) ) ).
27:26:  |-  (. u  =  v  ->.  ( E. y ( x  =  u  /\  y  =  u )  ->  E. y ( x  =  u  /\  y  =  v ) ) ).
28:27:  |-  (. u  =  v  ->.  A. x ( E. y ( x  =  u  /\  y  =  u )  ->  E. y ( x  =  u  /\  y  =  v ) ) ).
29:28:  |-  (. u  =  v  ->.  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y ( x  =  u  /\  y  =  v ) ) ).
30:29:  |-  ( u  =  v  ->  ( E. x E. y ( x  =  u  /\  y  =  u  )  ->  E. x E. y ( x  =  u  /\  y  =  v ) ) )
31:18,30:  |-  (. A. x x  =  y  ->.  ( u  =  v  ->  E. x E. y  ( x  =  u  /\  y  =  v ) ) ).
qed:31:  |-  ( A. x x  =  y  ->  ( u  =  v  ->  E. x E. y (  x  =  u  /\  y  =  v ) ) )
Assertion
Ref Expression
a9e2eqVD  |-  ( A. x  x  =  y  ->  ( u  =  v  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
Distinct variable groups:    x, u    y, u    x, v    y,
v

Proof of Theorem a9e2eqVD
StepHypRef Expression
1 idn1 28007 . . . . . 6  |-  (. A. x  x  =  y  ->.  A. x  x  =  y ).
2 a9ev 1663 . . . . . . . . . 10  |-  E. x  x  =  u
3 hba1 1794 . . . . . . . . . . . . . 14  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
4 sp 1755 . . . . . . . . . . . . . 14  |-  ( A. x A. x  x  =  y  ->  A. x  x  =  y )
53, 4impbii 181 . . . . . . . . . . . . 13  |-  ( A. x  x  =  y  <->  A. x A. x  x  =  y )
6 idn2 28056 . . . . . . . . . . . . . . . . 17  |-  (. A. x  x  =  y ,. x  =  u  ->.  x  =  u ).
7 sp 1755 . . . . . . . . . . . . . . . . . . 19  |-  ( A. x  x  =  y  ->  x  =  y )
81, 7e1_ 28070 . . . . . . . . . . . . . . . . . 18  |-  (. A. x  x  =  y  ->.  x  =  y ).
9 ax-8 1682 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  (
x  =  u  -> 
y  =  u ) )
109com12 29 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  u  ->  (
x  =  y  -> 
y  =  u ) )
116, 8, 10e21 28184 . . . . . . . . . . . . . . . . 17  |-  (. A. x  x  =  y ,. x  =  u  ->.  y  =  u ).
12 pm3.2 435 . . . . . . . . . . . . . . . . 17  |-  ( x  =  u  ->  (
y  =  u  -> 
( x  =  u  /\  y  =  u ) ) )
136, 11, 12e22 28114 . . . . . . . . . . . . . . . 16  |-  (. A. x  x  =  y ,. x  =  u  ->.  ( x  =  u  /\  y  =  u ) ).
1413in2 28048 . . . . . . . . . . . . . . 15  |-  (. A. x  x  =  y  ->.  ( x  =  u  -> 
( x  =  u  /\  y  =  u ) ) ).
1514in1 28004 . . . . . . . . . . . . . 14  |-  ( A. x  x  =  y  ->  ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) )
1615alimi 1565 . . . . . . . . . . . . 13  |-  ( A. x A. x  x  =  y  ->  A. x
( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) )
175, 16sylbi 188 . . . . . . . . . . . 12  |-  ( A. x  x  =  y  ->  A. x ( x  =  u  ->  (
x  =  u  /\  y  =  u )
) )
181, 17e1_ 28070 . . . . . . . . . . 11  |-  (. A. x  x  =  y  ->.  A. x ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) ).
19 exim 1581 . . . . . . . . . . 11  |-  ( A. x ( x  =  u  ->  ( x  =  u  /\  y  =  u ) )  -> 
( E. x  x  =  u  ->  E. x
( x  =  u  /\  y  =  u ) ) )
2018, 19e1_ 28070 . . . . . . . . . 10  |-  (. A. x  x  =  y  ->.  ( E. x  x  =  u  ->  E. x
( x  =  u  /\  y  =  u ) ) ).
21 pm2.27 37 . . . . . . . . . 10  |-  ( E. x  x  =  u  ->  ( ( E. x  x  =  u  ->  E. x ( x  =  u  /\  y  =  u ) )  ->  E. x ( x  =  u  /\  y  =  u ) ) )
222, 20, 21e01 28134 . . . . . . . . 9  |-  (. A. x  x  =  y  ->.  E. x ( x  =  u  /\  y  =  u ) ).
2322in1 28004 . . . . . . . 8  |-  ( A. x  x  =  y  ->  E. x ( x  =  u  /\  y  =  u ) )
2423a5i 1797 . . . . . . 7  |-  ( A. x  x  =  y  ->  A. x E. x
( x  =  u  /\  y  =  u ) )
251, 24e1_ 28070 . . . . . 6  |-  (. A. x  x  =  y  ->.  A. x E. x ( x  =  u  /\  y  =  u ) ).
26 ax10o 1993 . . . . . 6  |-  ( A. x  x  =  y  ->  ( A. x E. x ( x  =  u  /\  y  =  u )  ->  A. y E. x ( x  =  u  /\  y  =  u ) ) )
271, 25, 26e11 28131 . . . . 5  |-  (. A. x  x  =  y  ->.  A. y E. x ( x  =  u  /\  y  =  u ) ).
28 19.2 1666 . . . . 5  |-  ( A. y E. x ( x  =  u  /\  y  =  u )  ->  E. y E. x ( x  =  u  /\  y  =  u ) )
2927, 28e1_ 28070 . . . 4  |-  (. A. x  x  =  y  ->.  E. y E. x ( x  =  u  /\  y  =  u ) ).
30 excomim 1749 . . . 4  |-  ( E. y E. x ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  u ) )
3129, 30e1_ 28070 . . 3  |-  (. A. x  x  =  y  ->.  E. x E. y ( x  =  u  /\  y  =  u ) ).
32 idn1 28007 . . . . . . . . . . 11  |-  (. u  =  v  ->.  u  =  v ).
33 idn2 28056 . . . . . . . . . . . 12  |-  (. u  =  v ,. (
x  =  u  /\  y  =  u )  ->.  ( x  =  u  /\  y  =  u ) ).
34 simpr 448 . . . . . . . . . . . 12  |-  ( ( x  =  u  /\  y  =  u )  ->  y  =  u )
3533, 34e2 28074 . . . . . . . . . . 11  |-  (. u  =  v ,. (
x  =  u  /\  y  =  u )  ->.  y  =  u ).
36 equtrr 1690 . . . . . . . . . . 11  |-  ( u  =  v  ->  (
y  =  u  -> 
y  =  v ) )
3732, 35, 36e12 28178 . . . . . . . . . 10  |-  (. u  =  v ,. (
x  =  u  /\  y  =  u )  ->.  y  =  v ).
38 simpl 444 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  u )  ->  x  =  u )
3933, 38e2 28074 . . . . . . . . . 10  |-  (. u  =  v ,. (
x  =  u  /\  y  =  u )  ->.  x  =  u ).
40 pm3.21 436 . . . . . . . . . 10  |-  ( y  =  v  ->  (
x  =  u  -> 
( x  =  u  /\  y  =  v ) ) )
4137, 39, 40e22 28114 . . . . . . . . 9  |-  (. u  =  v ,. (
x  =  u  /\  y  =  u )  ->.  ( x  =  u  /\  y  =  v ) ).
4241in2 28048 . . . . . . . 8  |-  (. u  =  v  ->.  ( ( x  =  u  /\  y  =  u )  ->  (
x  =  u  /\  y  =  v )
) ).
4342gen11 28059 . . . . . . 7  |-  (. u  =  v  ->.  A. y ( ( x  =  u  /\  y  =  u )  ->  ( x  =  u  /\  y  =  v ) ) ).
44 exim 1581 . . . . . . 7  |-  ( A. y ( ( x  =  u  /\  y  =  u )  ->  (
x  =  u  /\  y  =  v )
)  ->  ( E. y ( x  =  u  /\  y  =  u )  ->  E. y
( x  =  u  /\  y  =  v ) ) )
4543, 44e1_ 28070 . . . . . 6  |-  (. u  =  v  ->.  ( E. y
( x  =  u  /\  y  =  u )  ->  E. y
( x  =  u  /\  y  =  v ) ) ).
4645gen11 28059 . . . . 5  |-  (. u  =  v  ->.  A. x ( E. y ( x  =  u  /\  y  =  u )  ->  E. y
( x  =  u  /\  y  =  v ) ) ).
47 exim 1581 . . . . 5  |-  ( A. x ( E. y
( x  =  u  /\  y  =  u )  ->  E. y
( x  =  u  /\  y  =  v ) )  ->  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
4846, 47e1_ 28070 . . . 4  |-  (. u  =  v  ->.  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y ( x  =  u  /\  y  =  v ) ) ).
4948in1 28004 . . 3  |-  ( u  =  v  ->  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
50 pm2.04 78 . . . 4  |-  ( ( u  =  v  -> 
( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y ( x  =  u  /\  y  =  v ) ) )  ->  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  (
u  =  v  ->  E. x E. y ( x  =  u  /\  y  =  v )
) ) )
5150com12 29 . . 3  |-  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  ( ( u  =  v  ->  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y ( x  =  u  /\  y  =  v ) ) )  ->  ( u  =  v  ->  E. x E. y ( x  =  u  /\  y  =  v ) ) ) )
5231, 49, 51e10 28137 . 2  |-  (. A. x  x  =  y  ->.  ( u  =  v  ->  E. x E. y ( x  =  u  /\  y  =  v )
) ).
5352in1 28004 1  |-  ( A. x  x  =  y  ->  ( u  =  v  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-vd1 28003  df-vd2 28012
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