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Theorem sbc3ang 3221
Description: Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbc3ang  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps  /\ 
ch )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) ) )

Proof of Theorem sbc3ang
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3166 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  /\  ps  /\  ch )  <->  [. A  /  x ]. ( ph  /\  ps  /\  ch ) ) )
2 dfsbcq2 3166 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 3166 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
4 dfsbcq2 3166 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ch  <->  [. A  /  x ]. ch ) )
52, 3, 43anbi123d 1255 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  /\  [
y  /  x ] ps  /\  [ y  /  x ] ch )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) ) )
6 sb3an 2142 . 2  |-  ( [ y  /  x ]
( ph  /\  ps  /\  ch )  <->  ( [ y  /  x ] ph  /\ 
[ y  /  x ] ps  /\  [ y  /  x ] ch ) )
71, 5, 6vtoclbg 3014 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps  /\ 
ch )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653   [wsb 1659    e. wcel 1726   [.wsbc 3163
This theorem is referenced by:  bnj156  29169  bnj206  29172  bnj976  29222  bnj121  29315  bnj130  29319  bnj581  29353  bnj1040  29415  cdlemkid3N  31804  cdlemkid4  31805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164
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