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Theorem ifmdc 3573
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 2240 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴 ∈ if(𝜑, 𝐵, 𝐶)))
21imbi1d 231 . . . 4 (𝑥 = 𝐴 → ((𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑)) ↔ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))))
3 df-if 3535 . . . . . 6 if(𝜑, 𝐵, 𝐶) = {𝑥 ∣ ((𝑥𝐵𝜑) ∨ (𝑥𝐶 ∧ ¬ 𝜑))}
43abeq2i 2288 . . . . 5 (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐶 ∧ ¬ 𝜑)))
5 simpr 110 . . . . . 6 ((𝑥𝐵𝜑) → 𝜑)
6 simpr 110 . . . . . 6 ((𝑥𝐶 ∧ ¬ 𝜑) → ¬ 𝜑)
75, 6orim12i 759 . . . . 5 (((𝑥𝐵𝜑) ∨ (𝑥𝐶 ∧ ¬ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
84, 7sylbi 121 . . . 4 (𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))
92, 8vtoclg 2797 . . 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 835 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
1210, 11sylibr 134 1 (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → DECID 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  ifcif 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-if 3535
This theorem is referenced by: (None)
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