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Theorem sbcang 3034
Description: Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)
Assertion
Ref Expression
sbcang  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
) )

Proof of Theorem sbcang
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2994 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  /\  ps )  <->  [. A  /  x ]. ( ph  /\  ps ) ) )
2 dfsbcq2 2994 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 2994 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3anbi12d 691 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  /\  [
y  /  x ] ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) ) )
5 sban 2009 . 2  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
61, 4, 5vtoclbg 2844 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   [wsb 1629    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  sbcabel  3068  csbunig  3835  csbxpg  4716  onfrALTlem5  28307  csbingVD  28660  onfrALTlem5VD  28661  onfrALTlem4VD  28662  csbxpgVD  28670  csbunigVD  28674
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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