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Theorem sbcal 3322
Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcal  |-  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph )
Distinct variable groups:    x, A    x, y
Allowed substitution hints:    ph( x, y)    A( y)

Proof of Theorem sbcal
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbcex 3280 . 2  |-  ( [. A  /  y ]. A. x ph  ->  A  e.  _V )
2 sbcex 3280 . . 3  |-  ( [. A  /  y ]. ph  ->  A  e.  _V )
32sps 1801 . 2  |-  ( A. x [. A  /  y ]. ph  ->  A  e.  _V )
4 dfsbcq2 3273 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] A. x ph  <->  [. A  / 
y ]. A. x ph ) )
5 dfsbcq2 3273 . . . 4  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
65albidv 1680 . . 3  |-  ( z  =  A  ->  ( A. x [ z  / 
y ] ph  <->  A. x [. A  /  y ]. ph ) )
7 sbal 2180 . . 3  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
84, 6, 7vtoclbg 3113 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph ) )
91, 3, 8pm5.21nii 353 1  |-  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1368    = wceq 1370   [wsb 1701    e. wcel 1757   _Vcvv 3054   [.wsbc 3270
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 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-v 3056  df-sbc 3271
This theorem is referenced by:  sbcabel  3359  sbcssg  3874  sbcfung  5525  sbcalf  29044  trsbc  31529  bnj89  31992  bnj538  32014  bnj110  32133  bnj611  32193  bnj1000  32216  bj-sbeq  32685  bj-sbceqgALT  32686
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