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Theorem sbcal 2963
Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
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 2920 . 2  |-  ( [. A  /  y ]. A. x ph  ->  A  e.  _V )
2 sbcex 2920 . . 3  |-  ( [. A  /  y ]. ph  ->  A  e.  _V )
32sps 1518 . 2  |-  ( A. x [. A  /  y ]. ph  ->  A  e.  _V )
4 dfsbcq2 2915 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] A. x ph  <->  [. A  / 
y ]. A. x ph ) )
5 dfsbcq2 2915 . . . 4  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
65albidv 1797 . . 3  |-  ( z  =  A  ->  ( A. x [ z  / 
y ] ph  <->  A. x [. A  /  y ]. ph ) )
7 sbal 1976 . . 3  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
84, 6, 7vtoclbg 2750 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph ) )
91, 3, 8pm5.21nii 694 1  |-  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1330    = wceq 1332    e. wcel 1481   [wsb 1736   _Vcvv 2689   [.wsbc 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2913
This theorem is referenced by:  sbcfung  5154
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