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Theorem ifbothdc 3357
 Description: A wff 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
ifbothdc.2
Assertion
Ref Expression
ifbothdc DECID

Proof of Theorem ifbothdc
StepHypRef Expression
1 iftrue 3336 . . . . . 6
21eqcomd 2045 . . . . 5
3 ifbothdc.1 . . . . 5
42, 3syl 14 . . . 4
54biimpcd 148 . . 3
653ad2ant1 925 . 2 DECID
7 iffalse 3339 . . . . . 6
87eqcomd 2045 . . . . 5
9 ifbothdc.2 . . . . 5
108, 9syl 14 . . . 4
1110biimpcd 148 . . 3
12113ad2ant2 926 . 2 DECID
13 exmiddc 744 . . 3 DECID
14133ad2ant3 927 . 2 DECID
156, 12, 14mpjaod 638 1 DECID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 98   wo 629  DECID wdc 742   w3a 885   wceq 1243  cif 3331 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-if 3332 This theorem is referenced by: (None)
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