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Theorem sbc3an 2889
Description: Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbc3an  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) )

Proof of Theorem sbc3an
StepHypRef Expression
1 df-3an 924 . . . 4  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
21sbcbii 2887 . . 3  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<-> 
[. A  /  x ]. ( ( ph  /\  ps )  /\  ch )
)
3 sbcan 2870 . . 3  |-  ( [. A  /  x ]. (
( ph  /\  ps )  /\  ch )  <->  ( [. A  /  x ]. ( ph  /\  ps )  /\  [. A  /  x ]. ch ) )
4 sbcan 2870 . . . 4  |-  ( [. A  /  x ]. ( ph  /\  ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) )
54anbi1i 446 . . 3  |-  ( (
[. A  /  x ]. ( ph  /\  ps )  /\  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch ) )
62, 3, 53bitri 204 . 2  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch )
)
7 df-3an 924 . 2  |-  ( (
[. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch )
)
86, 7bitr4i 185 1  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    /\ w3a 922   [.wsbc 2829
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-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-sbc 2830
This theorem is referenced by: (None)
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