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Theorem ralsng 3481
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 3479 . 2  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
2 ralsng.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32sbcieg 2871 . 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 1289    e. wcel 1438   A.wral 2359   [.wsbc 2840   {csn 3444
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-sbc 2841  df-sn 3450
This theorem is referenced by:  ralsn  3484  ralprg  3491  raltpg  3493  ralunsn  3639  iinxsng  3801  posng  4506  fimax2gtrilemstep  6606  iseqf1olemqk  9911  seq3f1olemstep  9918  fimaxre2  10645  nninfsellemdc  11785
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