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Theorem ifcldcd 3389
Description: Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
Hypotheses
Ref Expression
ifcldcd.a (𝜑𝐴𝐶)
ifcldcd.b (𝜑𝐵𝐶)
ifcldcd.dc (𝜑DECID 𝜓)
Assertion
Ref Expression
ifcldcd (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

Proof of Theorem ifcldcd
StepHypRef Expression
1 iftrue 3364 . . . 4 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
21adantl 271 . . 3 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴)
3 ifcldcd.a . . . 4 (𝜑𝐴𝐶)
43adantr 270 . . 3 ((𝜑𝜓) → 𝐴𝐶)
52, 4eqeltrd 2156 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
6 iffalse 3367 . . . 4 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
76adantl 271 . . 3 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵)
8 ifcldcd.b . . . 4 (𝜑𝐵𝐶)
98adantr 270 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝐵𝐶)
107, 9eqeltrd 2156 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
11 ifcldcd.dc . . 3 (𝜑DECID 𝜓)
12 df-dc 777 . . 3 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
1311, 12sylib 120 . 2 (𝜑 → (𝜓 ∨ ¬ 𝜓))
145, 10, 13mpjaodan 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 3359
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 2064
This theorem depends on definitions:  df-bi 115  df-dc 777  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-if 3360
This theorem is referenced by:  uzin2  10011
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