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Theorem ifan 3714
Description: Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
Assertion
Ref Expression
ifan  |-  if ( ( ph  /\  ps ) ,  A ,  B )  =  if ( ph ,  if ( ps ,  A ,  B ) ,  B
)

Proof of Theorem ifan
StepHypRef Expression
1 iftrue 3681 . . 3  |-  ( ph  ->  if ( ph ,  if ( ps ,  A ,  B ) ,  B
)  =  if ( ps ,  A ,  B ) )
2 ibar 491 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
32ifbid 3693 . . 3  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ( ph  /\ 
ps ) ,  A ,  B ) )
41, 3eqtr2d 2413 . 2  |-  ( ph  ->  if ( ( ph  /\ 
ps ) ,  A ,  B )  =  if ( ph ,  if ( ps ,  A ,  B ) ,  B
) )
5 simpl 444 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
65con3i 129 . . . 4  |-  ( -. 
ph  ->  -.  ( ph  /\ 
ps ) )
7 iffalse 3682 . . . 4  |-  ( -.  ( ph  /\  ps )  ->  if ( (
ph  /\  ps ) ,  A ,  B )  =  B )
86, 7syl 16 . . 3  |-  ( -. 
ph  ->  if ( (
ph  /\  ps ) ,  A ,  B )  =  B )
9 iffalse 3682 . . 3  |-  ( -. 
ph  ->  if ( ph ,  if ( ps ,  A ,  B ) ,  B )  =  B )
108, 9eqtr4d 2415 . 2  |-  ( -. 
ph  ->  if ( (
ph  /\  ps ) ,  A ,  B )  =  if ( ph ,  if ( ps ,  A ,  B ) ,  B ) )
114, 10pm2.61i 158 1  |-  if ( ( ph  /\  ps ) ,  A ,  B )  =  if ( ph ,  if ( ps ,  A ,  B ) ,  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649   ifcif 3675
This theorem is referenced by:  itg0  19531  iblre  19545  itgreval  19548  iblss  19556  iblss2  19557  itgle  19561  itgss  19563  itgeqa  19565  iblconst  19569  itgconst  19570  ibladdlem  19571  iblabslem  19579  iblabsr  19581  iblmulc2  19582  itgsplit  19587  itgcn  19594  itg2gt0cn  25953  ibladdnclem  25954  iblabsnclem  25961  iblmulc2nc  25963  bddiblnc  25968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-if 3676
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