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Theorem sbc3an 3026
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 980 . . . 4  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
21sbcbii 3024 . . 3  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<-> 
[. A  /  x ]. ( ( ph  /\  ps )  /\  ch )
)
3 sbcan 3007 . . 3  |-  ( [. A  /  x ]. (
( ph  /\  ps )  /\  ch )  <->  ( [. A  /  x ]. ( ph  /\  ps )  /\  [. A  /  x ]. ch ) )
4 sbcan 3007 . . . 4  |-  ( [. A  /  x ]. ( ph  /\  ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) )
54anbi1i 458 . . 3  |-  ( (
[. A  /  x ]. ( ph  /\  ps )  /\  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch ) )
62, 3, 53bitri 206 . 2  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch )
)
7 df-3an 980 . 2  |-  ( (
[. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch )
)
86, 7bitr4i 187 1  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    /\ w3a 978   [.wsbc 2964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965
This theorem is referenced by: (None)
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