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Theorem ifbothdc 3385
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 3364 . . . . . 6  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21eqcomd 2061 . . . . 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 152 . . 3  |-  ( ps 
->  ( ph  ->  th )
)
653ad2ant1 936 . 2  |-  ( ( ps  /\  ch  /\ DECID  ph )  ->  ( ph  ->  th )
)
7 iffalse 3367 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
87eqcomd 2061 . . . . 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 152 . . 3  |-  ( ch 
->  ( -.  ph  ->  th ) )
12113ad2ant2 937 . 2  |-  ( ( ps  /\  ch  /\ DECID  ph )  ->  ( -.  ph  ->  th ) )
13 exmiddc 755 . . 3  |-  (DECID  ph  ->  (
ph  \/  -.  ph )
)
14133ad2ant3 938 . 2  |-  ( ( ps  /\  ch  /\ DECID  ph )  ->  ( ph  \/  -.  ph ) )
156, 12, 14mpjaod 648 1  |-  ( ( ps  /\  ch  /\ DECID  ph )  ->  th )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 102    \/ wo 639  DECID wdc 753    /\ w3a 896    = wceq 1259   ifcif 3359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3an 898  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-if 3360
This theorem is referenced by: (None)
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