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Theorem ifbothdc 3399
Description: A wff 𝜃 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(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
ifbothdc.2 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
Assertion
Ref Expression
ifbothdc ((𝜓𝜒DECID 𝜑) → 𝜃)

Proof of Theorem ifbothdc
StepHypRef Expression
1 iftrue 3374 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21eqcomd 2088 . . . . 5 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
3 ifbothdc.1 . . . . 5 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
42, 3syl 14 . . . 4 (𝜑 → (𝜓𝜃))
54biimpcd 157 . . 3 (𝜓 → (𝜑𝜃))
653ad2ant1 960 . 2 ((𝜓𝜒DECID 𝜑) → (𝜑𝜃))
7 iffalse 3377 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
87eqcomd 2088 . . . . 5 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
9 ifbothdc.2 . . . . 5 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
108, 9syl 14 . . . 4 𝜑 → (𝜒𝜃))
1110biimpcd 157 . . 3 (𝜒 → (¬ 𝜑𝜃))
12113ad2ant2 961 . 2 ((𝜓𝜒DECID 𝜑) → (¬ 𝜑𝜃))
13 exmiddc 778 . . 3 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
14133ad2ant3 962 . 2 ((𝜓𝜒DECID 𝜑) → (𝜑 ∨ ¬ 𝜑))
156, 12, 14mpjaod 671 1 ((𝜓𝜒DECID 𝜑) → 𝜃)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 662  DECID wdc 776  w3a 920   = wceq 1285  ifcif 3369
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 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-if 3370
This theorem is referenced by: (None)
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