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Theorem sbcsng 3825
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
sbcsng  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x  e.  { A } ph ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem sbcsng
StepHypRef Expression
1 ralsns 3804 . 2  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
21bicomd 193 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x  e.  { A } ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   A.wral 2666   [.wsbc 3121   {csn 3774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-sbc 3122  df-sn 3780
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