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Theorem sbc3an 3376
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 976 . . . 4  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
21sbcbii 3373 . . 3  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<-> 
[. A  /  x ]. ( ( ph  /\  ps )  /\  ch )
)
3 sbcan 3356 . . 3  |-  ( [. A  /  x ]. (
( ph  /\  ps )  /\  ch )  <->  ( [. A  /  x ]. ( ph  /\  ps )  /\  [. A  /  x ]. ch ) )
4 sbcan 3356 . . . 4  |-  ( [. A  /  x ]. ( ph  /\  ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) )
54anbi1i 695 . . 3  |-  ( (
[. A  /  x ]. ( ph  /\  ps )  /\  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch ) )
62, 3, 53bitri 271 . 2  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch )
)
7 df-3an 976 . 2  |-  ( (
[. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch )
)
86, 7bitr4i 252 1  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 974   [.wsbc 3313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-v 3097  df-sbc 3314
This theorem is referenced by:  bnj156  33511  bnj206  33514  bnj976  33564  bnj121  33656  bnj130  33660  bnj581  33694  bnj1040  33756  cdlemkid3N  36393  cdlemkid4  36394
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