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Theorem ifcldcd 3512
Description: Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
Hypotheses
Ref Expression
ifcldcd.a  |-  ( ph  ->  A  e.  C )
ifcldcd.b  |-  ( ph  ->  B  e.  C )
ifcldcd.dc  |-  ( ph  -> DECID  ps )
Assertion
Ref Expression
ifcldcd  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )

Proof of Theorem ifcldcd
StepHypRef Expression
1 iftrue 3484 . . . 4  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
21adantl 275 . . 3  |-  ( (
ph  /\  ps )  ->  if ( ps ,  A ,  B )  =  A )
3 ifcldcd.a . . . 4  |-  ( ph  ->  A  e.  C )
43adantr 274 . . 3  |-  ( (
ph  /\  ps )  ->  A  e.  C )
52, 4eqeltrd 2217 . 2  |-  ( (
ph  /\  ps )  ->  if ( ps ,  A ,  B )  e.  C )
6 iffalse 3487 . . . 4  |-  ( -. 
ps  ->  if ( ps ,  A ,  B
)  =  B )
76adantl 275 . . 3  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  A ,  B )  =  B )
8 ifcldcd.b . . . 4  |-  ( ph  ->  B  e.  C )
98adantr 274 . . 3  |-  ( (
ph  /\  -.  ps )  ->  B  e.  C )
107, 9eqeltrd 2217 . 2  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  A ,  B )  e.  C )
11 ifcldcd.dc . . 3  |-  ( ph  -> DECID  ps )
12 df-dc 821 . . 3  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
1311, 12sylib 121 . 2  |-  ( ph  ->  ( ps  \/  -.  ps ) )
145, 10, 13mpjaodan 788 1  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 698  DECID wdc 820    = wceq 1332    e. wcel 1481   ifcif 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-dc 821  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-if 3480
This theorem is referenced by:  fimax2gtrilemstep  6802  fodjuf  7025  fodjum  7026  fodju0  7027  nnnninf  7031  mkvprop  7040  xaddf  9657  xaddval  9658  uzin2  10791  fsum3ser  11198  fsumsplit  11208  explecnv  11306  ennnfonelemp1  11955  nnsf  13374  peano4nninf  13375  nninfalllemn  13377  nninfsellemcl  13382  nninffeq  13391  dcapncf  13423
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