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Theorem 2sb5nd 28625
Description: Equivalence for double substitution 2sb5 2064 without distinct  x,  y requirement. 2sb5nd 28625 is derived from 2sb5ndVD 29002. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2sb5nd  |-  ( ( -.  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 2sb5nd
StepHypRef Expression
1 a9e2ndeq 28624 . 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 28617 . . . . . . . . 9  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
43exbii 1572 . . . . . . . 8  |-  ( E. y ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  <->  E. y
( ( x  =  u  /\  y  =  v )  /\  ph ) )
5 nfs1v 2058 . . . . . . . . . 10  |-  F/ y [ v  /  y ] ph
65nfsb 2061 . . . . . . . . 9  |-  F/ y [ u  /  x ] [ v  /  y ] ph
7619.41 1827 . . . . . . . 8  |-  ( E. y ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  <->  ( E. y ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) )
84, 7bitr3i 242 . . . . . . 7  |-  ( E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  ( E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) )
98exbii 1572 . . . . . 6  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  E. x ( E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) )
10 nfs1v 2058 . . . . . . 7  |-  F/ x [ u  /  x ] [ v  /  y ] ph
111019.41 1827 . . . . . 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 ) )
129, 11bitr2i 241 . . . . 5  |-  ( ( E. x E. y
( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ph ) )
1312anbi2i 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 )
) )
142, 13bitr3i 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 ) ) )
15 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 )
) ) )
1614, 15mpbir 200 . 2  |-  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  ( [ u  /  x ] [ v  / 
y ] ph  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph ) ) )
171, 16sylbi 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 1530   E.wex 1531    = wceq 1632   [wsb 1638
This theorem is referenced by:  2uasbanh  28626  2uasbanhVD  29003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-v 2803
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