ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifcldcd GIF version

Theorem ifcldcd 3405
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 3378 . . . 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 2159 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
6 iffalse 3381 . . . 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 2159 . 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 3373
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 3374
This theorem is referenced by:  fodjuomnilemf  6704  fodjuomnilemm  6705  fodjuomnilem0  6706  nnnninf  8644  uzin2  10246
  Copyright terms: Public domain W3C validator