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Theorem 2sb5ndALT 28709
Description: Equivalence for double substitution 2sb5 2051 without distinct  x,  y requirement. 2sb5nd 28326 is derived from 2sb5ndVD 28686. 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 28686. (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 a9e2ndeq 28325 . 2  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  <->  E. x E. y ( x  =  u  /\  y  =  v ) )
2 anabs5 784 . . . 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 28318 . . . . . . . . 9  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
43exbii 1569 . . . . . . . 8  |-  ( E. y ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  <->  E. y
( ( x  =  u  /\  y  =  v )  /\  ph ) )
5 hbs1 2044 . . . . . . . . . . . 12  |-  ( [ u  /  x ] [ v  /  y ] ph  ->  A. x [ u  /  x ] [ v  /  y ] ph )
6 id 19 . . . . . . . . . . . . 13  |-  ( A. x  x  =  y  ->  A. x  x  =  y )
7 ax10o 1892 . . . . . . . . . . . . 13  |-  ( A. x  x  =  y  ->  ( A. x [
u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph ) )
86, 7syl 15 . . . . . . . . . . . 12  |-  ( A. x  x  =  y  ->  ( A. x [
u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph ) )
9 pm3.33 568 . . . . . . . . . . . 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 644 . . . . . . . . . . 11  |-  ( A. x  x  =  y  ->  ( [ u  /  x ] [ v  / 
y ] ph  ->  A. y [ u  /  x ] [ v  / 
y ] ph )
)
11 hbs1 2044 . . . . . . . . . . . . . 14  |-  ( [ v  /  y ]
ph  ->  A. y [ v  /  y ] ph )
1211sbt 1973 . . . . . . . . . . . . 13  |-  [ u  /  x ] ( [ v  /  y ]
ph  ->  A. y [ v  /  y ] ph )
13 sbi1 2003 . . . . . . . . . . . . 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 8 . . . . . . . . . . . 12  |-  ( [ u  /  x ] [ v  /  y ] ph  ->  [ u  /  x ] A. y [ v  /  y ] ph )
15 id 19 . . . . . . . . . . . . . 14  |-  ( -. 
A. x  x  =  y  ->  -.  A. x  x  =  y )
16 ax10 1884 . . . . . . . . . . . . . . 15  |-  ( A. y  y  =  x  ->  A. x  x  =  y )
1716con3i 127 . . . . . . . . . . . . . 14  |-  ( -. 
A. x  x  =  y  ->  -.  A. y 
y  =  x )
1815, 17syl 15 . . . . . . . . . . . . 13  |-  ( -. 
A. x  x  =  y  ->  -.  A. y 
y  =  x )
19 sbal2 2073 . . . . . . . . . . . . 13  |-  ( -. 
A. y  y  =  x  ->  ( [
u  /  x ] A. y [ v  / 
y ] ph  <->  A. y [ u  /  x ] [ v  /  y ] ph ) )
2018, 19syl 15 . . . . . . . . . . . 12  |-  ( -. 
A. x  x  =  y  ->  ( [
u  /  x ] A. y [ v  / 
y ] ph  <->  A. y [ u  /  x ] [ v  /  y ] ph ) )
21 imbi2 314 . . . . . . . . . . . . 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 28335 . . . . . . . . . . . 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 644 . . . . . . . . . . 11  |-  ( -. 
A. x  x  =  y  ->  ( [
u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph ) )
2410, 23pm2.61i 156 . . . . . . . . . 10  |-  ( [ u  /  x ] [ v  /  y ] ph  ->  A. y [ u  /  x ] [ v  /  y ] ph )
2524nfi 1538 . . . . . . . . 9  |-  F/ y [ u  /  x ] [ v  /  y ] ph
262519.41 1815 . . . . . . . 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 242 . . . . . . 7  |-  ( E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  ( E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) )
2827exbii 1569 . . . . . 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 2045 . . . . . . 7  |-  F/ x [ u  /  x ] [ v  /  y ] ph
302919.41 1815 . . . . . 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 241 . . . . 5  |-  ( ( E. x E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ph ) )
3231anbi2i 675 . . . 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 242 . . 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 pm5.32 617 . . 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 200 . 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 187 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623   [wsb 1629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-v 2790
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