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Theorem ifmdc 3512
Description: If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.)
Assertion
Ref Expression
ifmdc (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → DECID 𝜑)

Proof of Theorem ifmdc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2203 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴 ∈ if(𝜑, 𝐵, 𝐶)))
21imbi1d 230 . . . 4 (𝑥 = 𝐴 → ((𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑)) ↔ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))))
3 df-if 3478 . . . . . 6 if(𝜑, 𝐵, 𝐶) = {𝑥 ∣ ((𝑥𝐵𝜑) ∨ (𝑥𝐶 ∧ ¬ 𝜑))}
43abeq2i 2251 . . . . 5 (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐶 ∧ ¬ 𝜑)))
5 simpr 109 . . . . . 6 ((𝑥𝐵𝜑) → 𝜑)
6 simpr 109 . . . . . 6 ((𝑥𝐶 ∧ ¬ 𝜑) → ¬ 𝜑)
75, 6orim12i 749 . . . . 5 (((𝑥𝐵𝜑) ∨ (𝑥𝐶 ∧ ¬ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
84, 7sylbi 120 . . . 4 (𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))
92, 8vtoclg 2749 . . 3 (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑)))
109pm2.43i 49 . 2 (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))
11 df-dc 821 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
1210, 11sylibr 133 1 (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → DECID 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 698  DECID wdc 820   = wceq 1332  wcel 1481  ifcif 3477
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-if 3478
This theorem is referenced by: (None)
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