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Theorem sbc3ang 3179
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 3124 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  /\  ps  /\  ch )  <->  [. A  /  x ]. ( ph  /\  ps  /\  ch ) ) )
2 dfsbcq2 3124 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 3124 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
4 dfsbcq2 3124 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ch  <->  [. A  /  x ]. ch ) )
52, 3, 43anbi123d 1254 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  /\  [
y  /  x ] ps  /\  [ y  /  x ] ch )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) ) )
6 sb3an 2119 . 2  |-  ( [ y  /  x ]
( ph  /\  ps  /\  ch )  <->  ( [ y  /  x ] ph  /\ 
[ y  /  x ] ps  /\  [ y  /  x ] ch ) )
71, 5, 6vtoclbg 2972 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 177    /\ w3a 936    = wceq 1649   [wsb 1655    e. wcel 1721   [.wsbc 3121
This theorem is referenced by:  bnj156  28801  bnj206  28804  bnj976  28854  bnj121  28947  bnj130  28951  bnj581  28985  bnj1040  29047  cdlemkid3N  31415  cdlemkid4  31416
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-sbc 3122
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