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Theorem 2sb5ndALT 37392
Description: Equivalence for double substitution 2sb5 2292 without distinct  x,  y requirement. 2sb5nd 36997 is derived from 2sb5ndVD 37370. 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 37370. (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 36996 . 2  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  <->  E. x E. y ( x  =  u  /\  y  =  v ) )
2 anabs5 826 . . . 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 36989 . . . . . . . . 9  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
43exbii 1726 . . . . . . . 8  |-  ( E. y ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  <->  E. y
( ( x  =  u  /\  y  =  v )  /\  ph ) )
5 hbs1 2285 . . . . . . . . . . . 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 2163 . . . . . . . . . . . . 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 pm3.33 595 . . . . . . . . . . . 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 676 . . . . . . . . . . 11  |-  ( A. x  x  =  y  ->  ( [ u  /  x ] [ v  / 
y ] ph  ->  A. y [ u  /  x ] [ v  / 
y ] ph )
)
11 hbs1 2285 . . . . . . . . . . . . . 14  |-  ( [ v  /  y ]
ph  ->  A. y [ v  /  y ] ph )
1211sbt 2268 . . . . . . . . . . . . 13  |-  [ u  /  x ] ( [ v  /  y ]
ph  ->  A. y [ v  /  y ] ph )
13 sbi1 2241 . . . . . . . . . . . . 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 2157 . . . . . . . . . . . . . . 15  |-  ( A. y  y  =  x  ->  A. x  x  =  y )
1716con3i 142 . . . . . . . . . . . . . 14  |-  ( -. 
A. x  x  =  y  ->  -.  A. y 
y  =  x )
1815, 17syl 17 . . . . . . . . . . . . 13  |-  ( -. 
A. x  x  =  y  ->  -.  A. y 
y  =  x )
19 sbal2 2310 . . . . . . . . . . . . 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 331 . . . . . . . . . . . . 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 )
) )
2221biimpac 494 . . . . . . . . . . . 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 676 . . . . . . . . . . 11  |-  ( -. 
A. x  x  =  y  ->  ( [
u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph ) )
2410, 23pm2.61i 169 . . . . . . . . . 10  |-  ( [ u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph )
2524nfi 1682 . . . . . . . . 9  |-  F/ y [ u  /  x ] [ v  /  y ] ph
262519.41 2070 . . . . . . . 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 259 . . . . . . 7  |-  ( E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  ( E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) )
2827exbii 1726 . . . . . 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 2286 . . . . . . 7  |-  F/ x [ u  /  x ] [ v  /  y ] ph
302919.41 2070 . . . . . 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 258 . . . . 5  |-  ( ( E. x E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ph ) )
3231anbi2i 708 . . . 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 259 . . 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 pm5.32 648 . . 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 214 . 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 200 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 189    \/ wo 375    /\ wa 376   A.wal 1450    = wceq 1452   E.wex 1671   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-ne 2643  df-v 3033
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator