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Theorem sbcangOLD 36933
Description: Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcan 3321 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcangOLD  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
) )

Proof of Theorem sbcangOLD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3281 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  /\  ps )  <->  [. A  /  x ]. ( ph  /\  ps ) ) )
2 dfsbcq2 3281 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 3281 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3anbi12d 722 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  /\  [
y  /  x ] ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) ) )
5 sban 2238 . 2  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
61, 4, 5vtoclbg 3119 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454   [wsb 1807    e. wcel 1897   [.wsbc 3278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-v 3058  df-sbc 3279
This theorem is referenced by:  csbunigOLD  37251  csbxpgOLD  37253  csbingVD  37320  onfrALTlem5VD  37321  onfrALTlem4VD  37322  csbxpgVD  37330  csbunigVD  37334
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