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Theorem sbcan 2869
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 2836 . 2  |-  ( [. A  /  x ]. ( ph  /\  ps )  ->  A  e.  _V )
2 sbcex 2836 . . 3  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
32adantl 271 . 2  |-  ( (
[. A  /  x ]. ph  /\  [. A  /  x ]. ps )  ->  A  e.  _V )
4 dfsbcq2 2831 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  /\  ps )  <->  [. A  /  x ]. ( ph  /\  ps ) ) )
5 dfsbcq2 2831 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
6 dfsbcq2 2831 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
75, 6anbi12d 457 . . 3  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  /\  [
y  /  x ] ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) ) )
8 sban 1874 . . 3  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
94, 7, 8vtoclbg 2672 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  /\  ps ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
) )
101, 3, 9pm5.21nii 653 1  |-  ( [. A  /  x ]. ( ph  /\  ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   [wsb 1689   _Vcvv 2614   [.wsbc 2828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-sbc 2829
This theorem is referenced by:  sbc3an  2888  difopab  4530  sbcfung  4995  sbcfng  5115  sbcfg  5116  f1od2  5938
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