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Theorem 2sb5ndALT 37329
Description: Equivalence for double substitution 2sb5 2272 without distinct  x,  y requirement. 2sb5nd 36927 is derived from 2sb5ndVD 37307. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in 2sb5ndVD 37307. (Contributed by Alan Sare, 19-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2sb5ndALT  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  -> 
( [ u  /  x ] [ v  / 
y ] ph  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph ) ) )
Distinct variable groups:    x, u    y, u    x, v    y,
v
Allowed substitution hints:    ph( x, y, v, u)

Proof of Theorem 2sb5ndALT
StepHypRef Expression
1 ax6e2ndeq 36926 . 2  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  <->  E. x E. y ( x  =  u  /\  y  =  v ) )
2 anabs5 818 . . . 4  |-  ( ( E. x E. y
( x  =  u  /\  y  =  v )  /\  ( E. x E. y ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) )  <->  ( E. x E. y ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) )
3 2pm13.193 36919 . . . . . . . . 9  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
43exbii 1718 . . . . . . . 8  |-  ( E. y ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  <->  E. y
( ( x  =  u  /\  y  =  v )  /\  ph ) )
5 hbs1 2265 . . . . . . . . . . . 12  |-  ( [ u  /  x ] [ v  /  y ] ph  ->  A. x [ u  /  x ] [ v  /  y ] ph )
6 id 22 . . . . . . . . . . . . 13  |-  ( A. x  x  =  y  ->  A. x  x  =  y )
7 axc11 2148 . . . . . . . . . . . . 13  |-  ( A. x  x  =  y  ->  ( A. x [
u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph ) )
86, 7syl 17 . . . . . . . . . . . 12  |-  ( A. x  x  =  y  ->  ( A. x [
u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph ) )
9 pm3.33 589 . . . . . . . . . . . 12  |-  ( ( ( [ u  /  x ] [ v  / 
y ] ph  ->  A. x [ u  /  x ] [ v  / 
y ] ph )  /\  ( A. x [
u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph ) )  -> 
( [ u  /  x ] [ v  / 
y ] ph  ->  A. y [ u  /  x ] [ v  / 
y ] ph )
)
105, 8, 9sylancr 669 . . . . . . . . . . 11  |-  ( A. x  x  =  y  ->  ( [ u  /  x ] [ v  / 
y ] ph  ->  A. y [ u  /  x ] [ v  / 
y ] ph )
)
11 hbs1 2265 . . . . . . . . . . . . . 14  |-  ( [ v  /  y ]
ph  ->  A. y [ v  /  y ] ph )
1211sbt 2248 . . . . . . . . . . . . 13  |-  [ u  /  x ] ( [ v  /  y ]
ph  ->  A. y [ v  /  y ] ph )
13 sbi1 2221 . . . . . . . . . . . . 13  |-  ( [ u  /  x ]
( [ v  / 
y ] ph  ->  A. y [ v  / 
y ] ph )  ->  ( [ u  /  x ] [ v  / 
y ] ph  ->  [ u  /  x ] A. y [ v  / 
y ] ph )
)
1412, 13ax-mp 5 . . . . . . . . . . . 12  |-  ( [ u  /  x ] [ v  /  y ] ph  ->  [ u  /  x ] A. y [ v  /  y ] ph )
15 id 22 . . . . . . . . . . . . . 14  |-  ( -. 
A. x  x  =  y  ->  -.  A. x  x  =  y )
16 axc11n 2143 . . . . . . . . . . . . . . 15  |-  ( A. y  y  =  x  ->  A. x  x  =  y )
1716con3i 141 . . . . . . . . . . . . . 14  |-  ( -. 
A. x  x  =  y  ->  -.  A. y 
y  =  x )
1815, 17syl 17 . . . . . . . . . . . . 13  |-  ( -. 
A. x  x  =  y  ->  -.  A. y 
y  =  x )
19 sbal2 2290 . . . . . . . . . . . . 13  |-  ( -. 
A. y  y  =  x  ->  ( [
u  /  x ] A. y [ v  / 
y ] ph  <->  A. y [ u  /  x ] [ v  /  y ] ph ) )
2018, 19syl 17 . . . . . . . . . . . 12  |-  ( -. 
A. x  x  =  y  ->  ( [
u  /  x ] A. y [ v  / 
y ] ph  <->  A. y [ u  /  x ] [ v  /  y ] ph ) )
21 imbi2 326 . . . . . . . . . . . . 13  |-  ( ( [ u  /  x ] A. y [ v  /  y ] ph  <->  A. y [ u  /  x ] [ v  / 
y ] ph )  ->  ( ( [ u  /  x ] [ v  /  y ] ph  ->  [ u  /  x ] A. y [ v  /  y ] ph ) 
<->  ( [ u  /  x ] [ v  / 
y ] ph  ->  A. y [ u  /  x ] [ v  / 
y ] ph )
) )
2221biimpa21 36935 . . . . . . . . . . . 12  |-  ( ( ( [ u  /  x ] [ v  / 
y ] ph  ->  [ u  /  x ] A. y [ v  / 
y ] ph )  /\  ( [ u  /  x ] A. y [ v  /  y ]
ph 
<-> 
A. y [ u  /  x ] [ v  /  y ] ph ) )  ->  ( [ u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph ) )
2314, 20, 22sylancr 669 . . . . . . . . . . 11  |-  ( -. 
A. x  x  =  y  ->  ( [
u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph ) )
2410, 23pm2.61i 168 . . . . . . . . . 10  |-  ( [ u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph )
2524nfi 1674 . . . . . . . . 9  |-  F/ y [ u  /  x ] [ v  /  y ] ph
262519.41 2051 . . . . . . . 8  |-  ( E. y ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  <->  ( E. y ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) )
274, 26bitr3i 255 . . . . . . 7  |-  ( E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  ( E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) )
2827exbii 1718 . . . . . 6  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  E. x ( E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) )
29 nfs1v 2266 . . . . . . 7  |-  F/ x [ u  /  x ] [ v  /  y ] ph
302919.41 2051 . . . . . 6  |-  ( E. x ( E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( E. x E. y ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) )
3128, 30bitr2i 254 . . . . 5  |-  ( ( E. x E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ph ) )
3231anbi2i 700 . . . 4  |-  ( ( E. x E. y
( x  =  u  /\  y  =  v )  /\  ( E. x E. y ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) )  <->  ( E. x E. y ( x  =  u  /\  y  =  v )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )
) )
332, 32bitr3i 255 . . 3  |-  ( ( E. x E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( E. x E. y ( x  =  u  /\  y  =  v )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph ) ) )
34 pm5.32 642 . . 3  |-  ( ( E. x E. y
( x  =  u  /\  y  =  v )  ->  ( [
u  /  x ] [ v  /  y ] ph  <->  E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ph ) ) )  <->  ( ( E. x E. y ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  <->  ( E. x E. y ( x  =  u  /\  y  =  v )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )
) ) )
3533, 34mpbir 213 . 2  |-  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  ( [ u  /  x ] [ v  / 
y ] ph  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph ) ) )
361, 35sylbi 199 1  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  -> 
( [ u  /  x ] [ v  / 
y ] ph  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371   A.wal 1442    = wceq 1444   E.wex 1663   [wsb 1797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-ne 2624  df-v 3047
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator