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Theorem sbcal 3029
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 2986 . 2  |-  ( [. A  /  y ]. A. x ph  ->  A  e.  _V )
2 sbcex 2986 . . 3  |-  ( [. A  /  y ]. ph  ->  A  e.  _V )
32sps 1548 . 2  |-  ( A. x [. A  /  y ]. ph  ->  A  e.  _V )
4 dfsbcq2 2980 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] A. x ph  <->  [. A  / 
y ]. A. x ph ) )
5 dfsbcq2 2980 . . . 4  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
65albidv 1835 . . 3  |-  ( z  =  A  ->  ( A. x [ z  / 
y ] ph  <->  A. x [. A  /  y ]. ph ) )
7 sbal 2012 . . 3  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
84, 6, 7vtoclbg 2813 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph ) )
91, 3, 8pm5.21nii 705 1  |-  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   A.wal 1362    = wceq 1364   [wsb 1773    e. wcel 2160   _Vcvv 2752   [.wsbc 2977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978
This theorem is referenced by:  sbcfung  5259
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