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Theorem ifbothdc 3427
 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 3402 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21eqcomd 2094 . . . . 5 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
3 ifbothdc.1 . . . . 5 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
42, 3syl 14 . . . 4 (𝜑 → (𝜓𝜃))
54biimpcd 158 . . 3 (𝜓 → (𝜑𝜃))
653ad2ant1 965 . 2 ((𝜓𝜒DECID 𝜑) → (𝜑𝜃))
7 iffalse 3405 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
87eqcomd 2094 . . . . 5 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
9 ifbothdc.2 . . . . 5 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
108, 9syl 14 . . . 4 𝜑 → (𝜒𝜃))
1110biimpcd 158 . . 3 (𝜒 → (¬ 𝜑𝜃))
12113ad2ant2 966 . 2 ((𝜓𝜒DECID 𝜑) → (¬ 𝜑𝜃))
13 exmiddc 783 . . 3 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
14133ad2ant3 967 . 2 ((𝜓𝜒DECID 𝜑) → (𝜑 ∨ ¬ 𝜑))
156, 12, 14mpjaod 674 1 ((𝜓𝜒DECID 𝜑) → 𝜃)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 665  DECID wdc 781   ∧ w3a 925   = wceq 1290  ifcif 3397 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-dc 782  df-3an 927  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-if 3398 This theorem is referenced by:  isumlessdc  10944
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