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Theorem ifcldcd 3355
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 3333 . . . 4 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
21adantl 262 . . 3 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴)
3 ifcldcd.a . . . 4 (𝜑𝐴𝐶)
43adantr 261 . . 3 ((𝜑𝜓) → 𝐴𝐶)
52, 4eqeltrd 2114 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
6 iffalse 3336 . . . 4 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
76adantl 262 . . 3 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵)
8 ifcldcd.b . . . 4 (𝜑𝐵𝐶)
98adantr 261 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝐵𝐶)
107, 9eqeltrd 2114 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
11 ifcldcd.dc . . 3 (𝜑DECID 𝜓)
12 df-dc 743 . . 3 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
1311, 12sylib 127 . 2 (𝜑 → (𝜓 ∨ ¬ 𝜓))
145, 10, 13mpjaodan 711 1 (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wo 629  DECID wdc 742   = wceq 1243  wcel 1393  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-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-if 3329
This theorem is referenced by:  uzin2  9440
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