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Theorem mo2icl 2743
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 1450 . . . . 5  |-  F/ x A. x ( ph  ->  x  =  A )
2 vex 2577 . . . . . . . 8  |-  x  e. 
_V
3 eleq1 2116 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  _V  <->  A  e.  _V ) )
42, 3mpbii 140 . . . . . . 7  |-  ( x  =  A  ->  A  e.  _V )
54imim2i 12 . . . . . 6  |-  ( (
ph  ->  x  =  A )  ->  ( ph  ->  A  e.  _V )
)
65sps 1446 . . . . 5  |-  ( A. x ( ph  ->  x  =  A )  -> 
( ph  ->  A  e. 
_V ) )
71, 6eximd 1519 . . . 4  |-  ( A. x ( ph  ->  x  =  A )  -> 
( E. x ph  ->  E. x  A  e. 
_V ) )
8 19.9v 1767 . . . 4  |-  ( E. x  A  e.  _V  <->  A  e.  _V )
97, 8syl6ib 154 . . 3  |-  ( A. x ( ph  ->  x  =  A )  -> 
( E. x ph  ->  A  e.  _V )
)
10 eqeq2 2065 . . . . . . . 8  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
1110imbi2d 223 . . . . . . 7  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
1211albidv 1721 . . . . . 6  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
1312imbi1d 224 . . . . 5  |-  ( y  =  A  ->  (
( A. x (
ph  ->  x  =  y )  ->  E* x ph )  <->  ( A. x
( ph  ->  x  =  A )  ->  E* x ph ) ) )
14 nfv 1437 . . . . . . 7  |-  F/ y
ph
1514mo2r 1968 . . . . . 6  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  E* x ph )
161519.23bi 1499 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  ->  E* x ph )
1713, 16vtoclg 2630 . . . 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 44 . 2  |-  ( A. x ( ph  ->  x  =  A )  -> 
( E. x ph  ->  E* x ph )
)
20 moabs 1965 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E* x ph ) )
2119, 20sylibr 141 1  |-  ( A. x ( ph  ->  x  =  A )  ->  E* x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   E*wmo 1917   _Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  invdisj  3787
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