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Theorem sbc3ang 3049
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 2994 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  /\  ps  /\  ch )  <->  [. A  /  x ]. ( ph  /\  ps  /\  ch ) ) )
2 dfsbcq2 2994 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 2994 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
4 dfsbcq2 2994 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ch  <->  [. A  /  x ]. ch ) )
52, 3, 43anbi123d 1252 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  /\  [
y  /  x ] ps  /\  [ y  /  x ] ch )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) ) )
6 sb3an 2010 . 2  |-  ( [ y  /  x ]
( ph  /\  ps  /\  ch )  <->  ( [ y  /  x ] ph  /\ 
[ y  /  x ] ps  /\  [ y  /  x ] ch ) )
71, 5, 6vtoclbg 2844 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 176    /\ w3a 934    = wceq 1623   [wsb 1629    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  bnj156  28756  bnj206  28759  bnj976  28809  bnj121  28902  bnj130  28906  bnj581  28940  bnj1040  29002  cdlemkid3N  31122  cdlemkid4  31123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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