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Theorem ifbothdc 3419
Description: A wff  th containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
Hypotheses
Ref Expression
ifbothdc.1  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th )
)
ifbothdc.2  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch  <->  th )
)
Assertion
Ref Expression
ifbothdc  |-  ( ( ps  /\  ch  /\ DECID  ph )  ->  th )

Proof of Theorem ifbothdc
StepHypRef Expression
1 iftrue 3394 . . . . . 6  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21eqcomd 2093 . . . . 5  |-  ( ph  ->  A  =  if (
ph ,  A ,  B ) )
3 ifbothdc.1 . . . . 5  |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th )
)
42, 3syl 14 . . . 4  |-  ( ph  ->  ( ps  <->  th )
)
54biimpcd 157 . . 3  |-  ( ps 
->  ( ph  ->  th )
)
653ad2ant1 964 . 2  |-  ( ( ps  /\  ch  /\ DECID  ph )  ->  ( ph  ->  th )
)
7 iffalse 3397 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
87eqcomd 2093 . . . . 5  |-  ( -. 
ph  ->  B  =  if ( ph ,  A ,  B ) )
9 ifbothdc.2 . . . . 5  |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch  <->  th )
)
108, 9syl 14 . . . 4  |-  ( -. 
ph  ->  ( ch  <->  th )
)
1110biimpcd 157 . . 3  |-  ( ch 
->  ( -.  ph  ->  th ) )
12113ad2ant2 965 . 2  |-  ( ( ps  /\  ch  /\ DECID  ph )  ->  ( -.  ph  ->  th ) )
13 exmiddc 782 . . 3  |-  (DECID  ph  ->  (
ph  \/  -.  ph )
)
14133ad2ant3 966 . 2  |-  ( ( ps  /\  ch  /\ DECID  ph )  ->  ( ph  \/  -.  ph ) )
156, 12, 14mpjaod 673 1  |-  ( ( ps  /\  ch  /\ DECID  ph )  ->  th )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 664  DECID wdc 780    /\ w3a 924    = wceq 1289   ifcif 3389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-if 3390
This theorem is referenced by: (None)
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