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Theorem ralsng 3559
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 3557 . 2  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
2 ralsng.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32sbcieg 2936 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
41, 3bitrd 187 1  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2414   [.wsbc 2904   {csn 3522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-sbc 2905  df-sn 3528
This theorem is referenced by:  ralsn  3562  ralprg  3569  raltpg  3571  ralunsn  3719  iinxsng  3881  posng  4606  fimax2gtrilemstep  6787  iseqf1olemqk  10260  seq3f1olemstep  10267  fimaxre2  10991  nninfsellemdc  13195
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