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Theorem ifcldadc 3395
Description: Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.)
Hypotheses
Ref Expression
ifcldadc.1 ((𝜑𝜓) → 𝐴𝐶)
ifcldadc.2 ((𝜑 ∧ ¬ 𝜓) → 𝐵𝐶)
ifcldadc.dc (𝜑DECID 𝜓)
Assertion
Ref Expression
ifcldadc (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

Proof of Theorem ifcldadc
StepHypRef Expression
1 iftrue 3373 . . . 4 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
21adantl 271 . . 3 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴)
3 ifcldadc.1 . . 3 ((𝜑𝜓) → 𝐴𝐶)
42, 3eqeltrd 2159 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
5 iffalse 3376 . . . 4 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
65adantl 271 . . 3 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵)
7 ifcldadc.2 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝐵𝐶)
86, 7eqeltrd 2159 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
9 ifcldadc.dc . . 3 (𝜑DECID 𝜓)
10 exmiddc 778 . . 3 (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓))
119, 10syl 14 . 2 (𝜑 → (𝜓 ∨ ¬ 𝜓))
124, 8, 11mpjaodan 745 1 (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  DECID wdc 776   = wceq 1285  wcel 1434  ifcif 3368
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-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-if 3369
This theorem is referenced by:  updjudhf  6573  eucalgval2  10660  lcmval  10670
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