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Theorem ralsnsg 4012
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
ralsnsg  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem ralsnsg
StepHypRef Expression
1 sbc6g 3314 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
2 df-ral 2801 . . 3  |-  ( A. x  e.  { A } ph  <->  A. x ( x  e.  { A }  ->  ph ) )
3 elsn 3994 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
43imbi1i 325 . . . 4  |-  ( ( x  e.  { A }  ->  ph )  <->  ( x  =  A  ->  ph )
)
54albii 1611 . . 3  |-  ( A. x ( x  e. 
{ A }  ->  ph )  <->  A. x ( x  =  A  ->  ph )
)
62, 5bitri 249 . 2  |-  ( A. x  e.  { A } ph  <->  A. x ( x  =  A  ->  ph )
)
71, 6syl6rbbr 264 1  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    = wceq 1370    e. wcel 1758   A.wral 2796   [.wsbc 3288   {csn 3980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-ral 2801  df-v 3074  df-sbc 3289  df-sn 3981
This theorem is referenced by:  ralsng  4015  sbcsngOLD  4036  ixpsnval  7371  ac6sfi  7662  rexfiuz  12948  prmind2  13887  finixpnum  28557
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