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Theorem mo2icl 2816
Description: Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
Assertion
Ref Expression
mo2icl  |-  ( A. x ( ph  ->  x  =  A )  ->  E* x ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem mo2icl
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfa1 1489 . . . . 5  |-  F/ x A. x ( ph  ->  x  =  A )
2 vex 2644 . . . . . . . 8  |-  x  e. 
_V
3 eleq1 2162 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  _V  <->  A  e.  _V ) )
42, 3mpbii 147 . . . . . . 7  |-  ( x  =  A  ->  A  e.  _V )
54imim2i 12 . . . . . 6  |-  ( (
ph  ->  x  =  A )  ->  ( ph  ->  A  e.  _V )
)
65sps 1485 . . . . 5  |-  ( A. x ( ph  ->  x  =  A )  -> 
( ph  ->  A  e. 
_V ) )
71, 6eximd 1559 . . . 4  |-  ( A. x ( ph  ->  x  =  A )  -> 
( E. x ph  ->  E. x  A  e. 
_V ) )
8 19.9v 1810 . . . 4  |-  ( E. x  A  e.  _V  <->  A  e.  _V )
97, 8syl6ib 160 . . 3  |-  ( A. x ( ph  ->  x  =  A )  -> 
( E. x ph  ->  A  e.  _V )
)
10 eqeq2 2109 . . . . . . . 8  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
1110imbi2d 229 . . . . . . 7  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
1211albidv 1763 . . . . . 6  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
1312imbi1d 230 . . . . 5  |-  ( y  =  A  ->  (
( A. x (
ph  ->  x  =  y )  ->  E* x ph )  <->  ( A. x
( ph  ->  x  =  A )  ->  E* x ph ) ) )
14 nfv 1476 . . . . . . 7  |-  F/ y
ph
1514mo2r 2012 . . . . . 6  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  E* x ph )
161519.23bi 1539 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  ->  E* x ph )
1713, 16vtoclg 2701 . . . 4  |-  ( A  e.  _V  ->  ( A. x ( ph  ->  x  =  A )  ->  E* x ph ) )
1817com12 30 . . 3  |-  ( A. x ( ph  ->  x  =  A )  -> 
( A  e.  _V  ->  E* x ph )
)
199, 18syld 45 . 2  |-  ( A. x ( ph  ->  x  =  A )  -> 
( E. x ph  ->  E* x ph )
)
20 moabs 2009 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E* x ph ) )
2119, 20sylibr 133 1  |-  ( A. x ( ph  ->  x  =  A )  ->  E* x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1297    = wceq 1299   E.wex 1436    e. wcel 1448   E*wmo 1961   _Vcvv 2641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643
This theorem is referenced by:  invdisj  3869
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