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Theorem sbcal 3038
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 3000 . 2  |-  ( [. A  /  y ]. A. x ph  ->  A  e.  _V )
2 sbcex 3000 . . 3  |-  ( [. A  /  y ]. ph  ->  A  e.  _V )
32sps 1739 . 2  |-  ( A. x [. A  /  y ]. ph  ->  A  e.  _V )
4 dfsbcq2 2994 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] A. x ph  <->  [. A  / 
y ]. A. x ph ) )
5 dfsbcq2 2994 . . . 4  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
65albidv 1611 . . 3  |-  ( z  =  A  ->  ( A. x [ z  / 
y ] ph  <->  A. x [. A  /  y ]. ph ) )
7 sbal 2066 . . 3  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
84, 6, 7vtoclbg 2844 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph ) )
91, 3, 8pm5.21nii 342 1  |-  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527    = wceq 1623   [wsb 1629    e. wcel 1684   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  sbcfun  27397  bnj110  28263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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