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Theorem sbcal 3379
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 3337 . 2  |-  ( [. A  /  y ]. A. x ph  ->  A  e.  _V )
2 sbcex 3337 . . 3  |-  ( [. A  /  y ]. ph  ->  A  e.  _V )
32sps 1866 . 2  |-  ( A. x [. A  /  y ]. ph  ->  A  e.  _V )
4 dfsbcq2 3330 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] A. x ph  <->  [. A  / 
y ]. A. x ph ) )
5 dfsbcq2 3330 . . . 4  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
65albidv 1714 . . 3  |-  ( z  =  A  ->  ( A. x [ z  / 
y ] ph  <->  A. x [. A  /  y ]. ph ) )
7 sbal 2207 . . 3  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
84, 6, 7vtoclbg 3168 . 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 1393    = wceq 1395   [wsb 1740    e. wcel 1819   _Vcvv 3109   [.wsbc 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-sbc 3328
This theorem is referenced by:  sbcabel  3412  sbcssg  3943  sbcfung  5617  sbcalf  30722  trsbc  33454  bnj89  33917  bnj538OLD  33940  bnj110  34059  bnj611  34119  bnj1000  34142  bj-sbeq  34611  bj-sbceqgALT  34612
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