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Theorem sbc3an 3357
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 967 . . . 4  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
21sbcbii 3354 . . 3  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<-> 
[. A  /  x ]. ( ( ph  /\  ps )  /\  ch )
)
3 sbcan 3337 . . 3  |-  ( [. A  /  x ]. (
( ph  /\  ps )  /\  ch )  <->  ( [. A  /  x ]. ( ph  /\  ps )  /\  [. A  /  x ]. ch ) )
4 sbcan 3337 . . . 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 967 . 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 965   [.wsbc 3294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3080  df-sbc 3295
This theorem is referenced by:  bnj156  32052  bnj206  32055  bnj976  32104  bnj121  32196  bnj130  32200  bnj581  32234  bnj1040  32296  cdlemkid3N  34916  cdlemkid4  34917
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