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Theorem sbcan 3146
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 3113 . 2  |-  ( [. A  /  x ]. ( ph  /\  ps )  ->  A  e.  _V )
2 sbcex 3113 . . 3  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
32adantl 453 . 2  |-  ( (
[. A  /  x ]. ph  /\  [. A  /  x ]. ps )  ->  A  e.  _V )
4 dfsbcq2 3107 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  /\  ps )  <->  [. A  /  x ]. ( ph  /\  ps ) ) )
5 dfsbcq2 3107 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
6 dfsbcq2 3107 . . . 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 2102 . . 3  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
94, 7, 8vtoclbg 2955 . 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 1649   [wsb 1655    e. wcel 1717   _Vcvv 2899   [.wsbc 3104
This theorem is referenced by:  difopab  4946  sbiota1  27303  sbcfun  27655  onfrALTlem4  27972  bnj976  28486  bnj110  28567  bnj1040  28679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105
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