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Theorem ralsng 3452
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
ralsng.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ralsng  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  ps ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem ralsng
StepHypRef Expression
1 ralsns 3450 . 2  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
2 ralsng.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32sbcieg 2856 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
41, 3bitrd 186 1  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2353   [.wsbc 2825   {csn 3417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-sbc 2826  df-sn 3423
This theorem is referenced by:  ralsn  3455  ralprg  3462  raltpg  3464  ralunsn  3610  iinxsng  3772  posng  4459  fimaxre2  10294
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