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Theorem sbcalgOLD 3340
Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) Obsolete as of 17-Aug-2018. Use sbcal 3339 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcalgOLD  |-  ( A  e.  V  ->  ( [. 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)    V( x, y)

Proof of Theorem sbcalgOLD
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3290 . 2  |-  ( z  =  A  ->  ( [ z  /  y ] A. x ph  <->  [. A  / 
y ]. A. x ph ) )
2 dfsbcq2 3290 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
32albidv 1680 . 2  |-  ( z  =  A  ->  ( A. x [ z  / 
y ] ph  <->  A. x [. A  /  y ]. ph ) )
4 sbal 2181 . 2  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
51, 3, 4vtoclbg 3130 1  |-  ( A  e.  V  ->  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    = wceq 1370   [wsb 1702    e. wcel 1758   [.wsbc 3287
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 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
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 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3073  df-sbc 3288
This theorem is referenced by:  sbcssOLD  31552  trsbcVD  31916  sbcssgVD  31922
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