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Theorem ifbothdc 3354
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 3333 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21eqcomd 2045 . . . . 5 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
3 ifbothdc.1 . . . . 5 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
42, 3syl 14 . . . 4 (𝜑 → (𝜓𝜃))
54biimpcd 148 . . 3 (𝜓 → (𝜑𝜃))
653ad2ant1 925 . 2 ((𝜓𝜒DECID 𝜑) → (𝜑𝜃))
7 iffalse 3336 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
87eqcomd 2045 . . . . 5 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
9 ifbothdc.2 . . . . 5 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
108, 9syl 14 . . . 4 𝜑 → (𝜒𝜃))
1110biimpcd 148 . . 3 (𝜒 → (¬ 𝜑𝜃))
12113ad2ant2 926 . 2 ((𝜓𝜒DECID 𝜑) → (¬ 𝜑𝜃))
13 exmiddc 744 . . 3 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
14133ad2ant3 927 . 2 ((𝜓𝜒DECID 𝜑) → (𝜑 ∨ ¬ 𝜑))
156, 12, 14mpjaod 638 1 ((𝜓𝜒DECID 𝜑) → 𝜃)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  wo 629  DECID wdc 742  w3a 885   = wceq 1243  ifcif 3328
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 3329
This theorem is referenced by: (None)
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