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Theorem 2mos 2372
Description: Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
Hypothesis
Ref Expression
2mos.1  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
2mos  |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
Distinct variable groups:    z, w, ph    x, y, ps    x, z, w, y
Allowed substitution hints:    ph( x, y)    ps( z, w)

Proof of Theorem 2mos
StepHypRef Expression
1 2mo 2370 . 2  |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\ 
[ z  /  x ] [ w  /  y ] ph )  ->  (
x  =  z  /\  y  =  w )
) )
2 nfv 1712 . . . . . . 7  |-  F/ x ps
3 2mos.1 . . . . . . . 8  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
43sbiedv 2154 . . . . . . 7  |-  ( x  =  z  ->  ( [ w  /  y ] ph  <->  ps ) )
52, 4sbie 2151 . . . . . 6  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  ps )
65anbi2i 692 . . . . 5  |-  ( (
ph  /\  [ z  /  x ] [ w  /  y ] ph ) 
<->  ( ph  /\  ps ) )
76imbi1i 323 . . . 4  |-  ( ( ( ph  /\  [
z  /  x ] [ w  /  y ] ph )  ->  (
x  =  z  /\  y  =  w )
)  <->  ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
872albii 1646 . . 3  |-  ( A. z A. w ( (
ph  /\  [ z  /  x ] [ w  /  y ] ph )  ->  ( x  =  z  /\  y  =  w ) )  <->  A. z A. w ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
982albii 1646 . 2  |-  ( A. x A. y A. z A. w ( ( ph  /\ 
[ z  /  x ] [ w  /  y ] ph )  ->  (
x  =  z  /\  y  =  w )
)  <->  A. x A. y A. z A. w ( ( ph  /\  ps )  ->  ( x  =  z  /\  y  =  w ) ) )
101, 9bitri 249 1  |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396   E.wex 1617   [wsb 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289
This theorem is referenced by: (None)
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