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Theorem sbccsb 3810
Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbccsb  |-  ( [. A  /  x ]. ph  <->  y  e.  [_ A  /  x ]_ { y  |  ph } )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)

Proof of Theorem sbccsb
StepHypRef Expression
1 abid 2441 . . 3  |-  ( y  e.  { y  | 
ph }  <->  ph )
21sbcbii 3354 . 2  |-  ( [. A  /  x ]. y  e.  { y  |  ph } 
<-> 
[. A  /  x ]. ph )
3 sbcel2 3792 . 2  |-  ( [. A  /  x ]. y  e.  { y  |  ph } 
<->  y  e.  [_ A  /  x ]_ { y  |  ph } )
42, 3bitr3i 251 1  |-  ( [. A  /  x ]. ph  <->  y  e.  [_ A  /  x ]_ { y  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1758   {cab 2439   [.wsbc 3294   [_csb 3396
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-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3747
This theorem is referenced by: (None)
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