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Theorem ralsns 3846
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
ralsns  |-  ( 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 ralsns
StepHypRef Expression
1 sbc6g 3188 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
2 df-ral 2712 . . 3  |-  ( A. x  e.  { A } ph  <->  A. x ( x  e.  { A }  ->  ph ) )
3 elsn 3831 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
43imbi1i 317 . . . 4  |-  ( ( x  e.  { A }  ->  ph )  <->  ( x  =  A  ->  ph )
)
54albii 1576 . . 3  |-  ( A. x ( x  e. 
{ A }  ->  ph )  <->  A. x ( x  =  A  ->  ph )
)
62, 5bitri 242 . 2  |-  ( A. x  e.  { A } ph  <->  A. x ( x  =  A  ->  ph )
)
71, 6syl6rbbr 257 1  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2707   [.wsbc 3163   {csn 3816
This theorem is referenced by:  ralsng  3848  sbcsng  3867  rabrsn  3876  ac6sfi  7353  rexfiuz  12153  prmind2  13092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-sbc 3164  df-sn 3822
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