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Theorem ralsnsg 4048
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised 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 3350 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
2 df-ral 2809 . . 3  |-  ( A. x  e.  { A } ph  <->  A. x ( x  e.  { A }  ->  ph ) )
3 elsn 4030 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
43imbi1i 323 . . . 4  |-  ( ( x  e.  { A }  ->  ph )  <->  ( x  =  A  ->  ph )
)
54albii 1645 . . 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 1396    = wceq 1398    e. wcel 1823   A.wral 2804   [.wsbc 3324   {csn 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-ral 2809  df-v 3108  df-sbc 3325  df-sn 4017
This theorem is referenced by:  ralsng  4051  ixpsnval  7465  ac6sfi  7756  rexfiuz  13265  prmind2  14315  finixpnum  30281
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