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Theorem sbc3an 3313
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 1009 . . . 4  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
21sbcbii 3311 . . 3  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<-> 
[. A  /  x ]. ( ( ph  /\  ps )  /\  ch )
)
3 sbcan 3298 . . 3  |-  ( [. A  /  x ]. (
( ph  /\  ps )  /\  ch )  <->  ( [. A  /  x ]. ( ph  /\  ps )  /\  [. A  /  x ]. ch ) )
4 sbcan 3298 . . . 4  |-  ( [. A  /  x ]. ( ph  /\  ps )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps ) )
54anbi1i 709 . . 3  |-  ( (
[. A  /  x ]. ( ph  /\  ps )  /\  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch ) )
62, 3, 53bitri 279 . 2  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch )
)
7 df-3an 1009 . 2  |-  ( (
[. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )  /\  [. A  /  x ]. ch )
)
86, 7bitr4i 260 1  |-  ( [. A  /  x ]. ( ph  /\  ps  /\  ch ) 
<->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007   [.wsbc 3255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033  df-sbc 3256
This theorem is referenced by:  bnj156  29608  bnj206  29611  bnj976  29661  bnj121  29753  bnj130  29757  bnj581  29791  bnj1040  29853  csbwrecsg  31798  topdifinffinlem  31820  rdgeqoa  31843  cdlemkid3N  34571  cdlemkid4  34572
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