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Theorem sbcan 3195
Description: Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
Assertion
Ref Expression
sbcan  |-  ( [. A  /  x ]. ( ph  /\  ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) )

Proof of Theorem sbcan
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sbcex 3162 . 2  |-  ( [. A  /  x ]. ( ph  /\  ps )  ->  A  e.  _V )
2 sbcex 3162 . . 3  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
32adantl 453 . 2  |-  ( (
[. A  /  x ]. ph  /\  [. A  /  x ]. ps )  ->  A  e.  _V )
4 dfsbcq2 3156 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  /\  ps )  <->  [. A  /  x ]. ( ph  /\  ps ) ) )
5 dfsbcq2 3156 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
6 dfsbcq2 3156 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
75, 6anbi12d 692 . . 3  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  /\  [
y  /  x ] ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) ) )
8 sban 2143 . . 3  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
94, 7, 8vtoclbg 3004 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  /\  ps ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
) )
101, 3, 9pm5.21nii 343 1  |-  ( [. A  /  x ]. ( ph  /\  ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652   [wsb 1658    e. wcel 1725   _Vcvv 2948   [.wsbc 3153
This theorem is referenced by:  difopab  4998  sbiota1  27592  sbcfun  27944  onfrALTlem4  28556  bnj976  29075  bnj110  29156  bnj1040  29268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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