MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbccsb2gOLD Structured version   Unicode version

Theorem sbccsb2gOLD 3838
Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) Obsolete as of 18-Aug-2018. Use sbccsb2 3837 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbccsb2gOLD  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph } ) )

Proof of Theorem sbccsb2gOLD
StepHypRef Expression
1 abid 2430 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21sbcbii 3373 . 2  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[. A  /  x ]. ph )
3 sbcel12gOLD 3810 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  { x  |  ph }  <->  [_ A  /  x ]_ x  e.  [_ A  /  x ]_ {
x  |  ph }
) )
4 csbvarg 3834 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ x  =  A )
54eleq1d 2512 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ x  e.  [_ A  /  x ]_ { x  |  ph }  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
) )
63, 5bitrd 253 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  { x  |  ph }  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
) )
72, 6syl5bbr 259 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1804   {cab 2428   [.wsbc 3313   [_csb 3420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-sbc 3314  df-csb 3421
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator