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Theorem sbccsb 3849
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 2454 . . 3  |-  ( y  e.  { y  | 
ph }  <->  ph )
21sbcbii 3391 . 2  |-  ( [. A  /  x ]. y  e.  { y  |  ph } 
<-> 
[. A  /  x ]. ph )
3 sbcel2 3831 . 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 1767   {cab 2452   [.wsbc 3331   [_csb 3435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786
This theorem is referenced by: (None)
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