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Theorem ax6e2eqVD 34089
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 33721) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2eq 33705 is ax6e2eqVD 34089 without virtual deductions and was automatically derived from ax6e2eqVD 34089. (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
ax6e2eqVD  |-  ( 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 ax6e2eqVD
StepHypRef Expression
1 idn1 33726 . . . . . 6  |-  (. A. x  x  =  y  ->.  A. x  x  =  y ).
2 ax6ev 1767 . . . . . . . . . 10  |-  E. x  x  =  u
3 hba1 1914 . . . . . . . . . . . . . 14  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
4 sp 1877 . . . . . . . . . . . . . 14  |-  ( A. x A. x  x  =  y  ->  A. x  x  =  y )
53, 4impbii 188 . . . . . . . . . . . . 13  |-  ( A. x  x  =  y  <->  A. x A. x  x  =  y )
6 idn2 33774 . . . . . . . . . . . . . . . . 17  |-  (. A. x  x  =  y ,. x  =  u  ->.  x  =  u ).
7 sp 1877 . . . . . . . . . . . . . . . . . . 19  |-  ( A. x  x  =  y  ->  x  =  y )
81, 7e1a 33788 . . . . . . . . . . . . . . . . . 18  |-  (. A. x  x  =  y  ->.  x  =  y ).
9 ax-7 1808 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  (
x  =  u  -> 
y  =  u ) )
109com12 31 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  u  ->  (
x  =  y  -> 
y  =  u ) )
116, 8, 10e21 33902 . . . . . . . . . . . . . . . . 17  |-  (. A. x  x  =  y ,. x  =  u  ->.  y  =  u ).
12 pm3.2 445 . . . . . . . . . . . . . . . . 17  |-  ( x  =  u  ->  (
y  =  u  -> 
( x  =  u  /\  y  =  u ) ) )
136, 11, 12e22 33832 . . . . . . . . . . . . . . . 16  |-  (. A. x  x  =  y ,. x  =  u  ->.  ( x  =  u  /\  y  =  u ) ).
1413in2 33766 . . . . . . . . . . . . . . 15  |-  (. A. x  x  =  y  ->.  ( x  =  u  -> 
( x  =  u  /\  y  =  u ) ) ).
1514in1 33723 . . . . . . . . . . . . . 14  |-  ( A. x  x  =  y  ->  ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) )
1615alimi 1648 . . . . . . . . . . . . 13  |-  ( A. x A. x  x  =  y  ->  A. x
( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) )
175, 16sylbi 195 . . . . . . . . . . . 12  |-  ( A. x  x  =  y  ->  A. x ( x  =  u  ->  (
x  =  u  /\  y  =  u )
) )
181, 17e1a 33788 . . . . . . . . . . 11  |-  (. A. x  x  =  y  ->.  A. x ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) ).
19 exim 1669 . . . . . . . . . . 11  |-  ( A. x ( x  =  u  ->  ( x  =  u  /\  y  =  u ) )  -> 
( E. x  x  =  u  ->  E. x
( x  =  u  /\  y  =  u ) ) )
2018, 19e1a 33788 . . . . . . . . . 10  |-  (. A. x  x  =  y  ->.  ( E. x  x  =  u  ->  E. x
( x  =  u  /\  y  =  u ) ) ).
21 pm2.27 39 . . . . . . . . . 10  |-  ( E. x  x  =  u  ->  ( ( E. x  x  =  u  ->  E. x ( x  =  u  /\  y  =  u ) )  ->  E. x ( x  =  u  /\  y  =  u ) ) )
222, 20, 21e01 33852 . . . . . . . . 9  |-  (. A. x  x  =  y  ->.  E. x ( x  =  u  /\  y  =  u ) ).
2322in1 33723 . . . . . . . 8  |-  ( A. x  x  =  y  ->  E. x ( x  =  u  /\  y  =  u ) )
2423axc4i 1916 . . . . . . 7  |-  ( A. x  x  =  y  ->  A. x E. x
( x  =  u  /\  y  =  u ) )
251, 24e1a 33788 . . . . . 6  |-  (. A. x  x  =  y  ->.  A. x E. x ( x  =  u  /\  y  =  u ) ).
26 axc11 2070 . . . . . 6  |-  ( A. x  x  =  y  ->  ( A. x E. x ( x  =  u  /\  y  =  u )  ->  A. y E. x ( x  =  u  /\  y  =  u ) ) )
271, 25, 26e11 33849 . . . . 5  |-  (. A. x  x  =  y  ->.  A. y E. x ( x  =  u  /\  y  =  u ) ).
28 19.2 1769 . . . . 5  |-  ( A. y E. x ( x  =  u  /\  y  =  u )  ->  E. y E. x ( x  =  u  /\  y  =  u ) )
2927, 28e1a 33788 . . . 4  |-  (. A. x  x  =  y  ->.  E. y E. x ( x  =  u  /\  y  =  u ) ).
30 excomim 1868 . . . 4  |-  ( E. y E. x ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  u ) )
3129, 30e1a 33788 . . 3  |-  (. A. x  x  =  y  ->.  E. x E. y ( x  =  u  /\  y  =  u ) ).
32 idn1 33726 . . . . . . . . . . 11  |-  (. u  =  v  ->.  u  =  v ).
33 idn2 33774 . . . . . . . . . . . 12  |-  (. u  =  v ,. (
x  =  u  /\  y  =  u )  ->.  ( x  =  u  /\  y  =  u ) ).
34 simpr 459 . . . . . . . . . . . 12  |-  ( ( x  =  u  /\  y  =  u )  ->  y  =  u )
3533, 34e2 33792 . . . . . . . . . . 11  |-  (. u  =  v ,. (
x  =  u  /\  y  =  u )  ->.  y  =  u ).
36 equtrr 1815 . . . . . . . . . . 11  |-  ( u  =  v  ->  (
y  =  u  -> 
y  =  v ) )
3732, 35, 36e12 33896 . . . . . . . . . 10  |-  (. u  =  v ,. (
x  =  u  /\  y  =  u )  ->.  y  =  v ).
38 simpl 455 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  u )  ->  x  =  u )
3933, 38e2 33792 . . . . . . . . . 10  |-  (. u  =  v ,. (
x  =  u  /\  y  =  u )  ->.  x  =  u ).
40 pm3.21 446 . . . . . . . . . 10  |-  ( y  =  v  ->  (
x  =  u  -> 
( x  =  u  /\  y  =  v ) ) )
4137, 39, 40e22 33832 . . . . . . . . 9  |-  (. u  =  v ,. (
x  =  u  /\  y  =  u )  ->.  ( x  =  u  /\  y  =  v ) ).
4241in2 33766 . . . . . . . 8  |-  (. u  =  v  ->.  ( ( x  =  u  /\  y  =  u )  ->  (
x  =  u  /\  y  =  v )
) ).
4342gen11 33777 . . . . . . 7  |-  (. u  =  v  ->.  A. y ( ( x  =  u  /\  y  =  u )  ->  ( x  =  u  /\  y  =  v ) ) ).
44 exim 1669 . . . . . . 7  |-  ( A. y ( ( x  =  u  /\  y  =  u )  ->  (
x  =  u  /\  y  =  v )
)  ->  ( E. y ( x  =  u  /\  y  =  u )  ->  E. y
( x  =  u  /\  y  =  v ) ) )
4543, 44e1a 33788 . . . . . 6  |-  (. u  =  v  ->.  ( E. y
( x  =  u  /\  y  =  u )  ->  E. y
( x  =  u  /\  y  =  v ) ) ).
4645gen11 33777 . . . . 5  |-  (. u  =  v  ->.  A. x ( E. y ( x  =  u  /\  y  =  u )  ->  E. y
( x  =  u  /\  y  =  v ) ) ).
47 exim 1669 . . . . 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, 47e1a 33788 . . . 4  |-  (. u  =  v  ->.  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y ( x  =  u  /\  y  =  v ) ) ).
4948in1 33723 . . 3  |-  ( u  =  v  ->  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
50 pm2.04 82 . . . 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 31 . . 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 33855 . 2  |-  (. A. x  x  =  y  ->.  ( u  =  v  ->  E. x E. y ( x  =  u  /\  y  =  v )
) ).
5352in1 33723 1  |-  ( A. x  x  =  y  ->  ( u  =  v  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1397    = wceq 1399   E.wex 1627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1628  df-nf 1632  df-vd1 33722  df-vd2 33730
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator