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Theorem ifmdc 3448
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 2157 . . . . 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 3414 . . . . . 6 if(𝜑, 𝐵, 𝐶) = {𝑥 ∣ ((𝑥𝐵𝜑) ∨ (𝑥𝐶 ∧ ¬ 𝜑))}
43abeq2i 2205 . . . . 5 (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐶 ∧ ¬ 𝜑)))
5 simpr 109 . . . . . 6 ((𝑥𝐵𝜑) → 𝜑)
6 simpr 109 . . . . . 6 ((𝑥𝐶 ∧ ¬ 𝜑) → ¬ 𝜑)
75, 6orim12i 714 . . . . 5 (((𝑥𝐵𝜑) ∨ (𝑥𝐶 ∧ ¬ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
84, 7sylbi 120 . . . 4 (𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))
92, 8vtoclg 2693 . . 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 784 . 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 667  DECID wdc 783   = wceq 1296  wcel 1445  ifcif 3413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-dc 784  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-if 3414
This theorem is referenced by: (None)
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