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Theorem ifcldcd 3507
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 3479 . . . 4 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
21adantl 275 . . 3 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴)
3 ifcldcd.a . . . 4 (𝜑𝐴𝐶)
43adantr 274 . . 3 ((𝜑𝜓) → 𝐴𝐶)
52, 4eqeltrd 2216 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
6 iffalse 3482 . . . 4 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
76adantl 275 . . 3 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵)
8 ifcldcd.b . . . 4 (𝜑𝐵𝐶)
98adantr 274 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝐵𝐶)
107, 9eqeltrd 2216 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
11 ifcldcd.dc . . 3 (𝜑DECID 𝜓)
12 df-dc 820 . . 3 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
1311, 12sylib 121 . 2 (𝜑 → (𝜓 ∨ ¬ 𝜓))
145, 10, 13mpjaodan 787 1 (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697  DECID wdc 819   = wceq 1331  wcel 1480  ifcif 3474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-if 3475
This theorem is referenced by:  fimax2gtrilemstep  6794  fodjuf  7017  fodjum  7018  fodju0  7019  nnnninf  7023  mkvprop  7032  xaddf  9639  xaddval  9640  uzin2  10771  fsum3ser  11178  fsumsplit  11188  explecnv  11286  ennnfonelemp1  11930  nnsf  13285  peano4nninf  13286  nninfalllemn  13288  nninfsellemcl  13293  nninffeq  13302
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