Theorem ifmdc 3558

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.

Step	Hyp	Ref	Expression
1		eleq1 2229	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴 ∈ if(𝜑, 𝐵, 𝐶)))
2	1	imbi1d 230	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑)) ↔ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))))
3		df-if 3521	. . . . . 6 ⊢ if(𝜑, 𝐵, 𝐶) = {𝑥 ∣ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑))}
4	3	abeq2i 2277	. . . . 5 ⊢ (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑)))
5		simpr 109	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝜑)
6		simpr 109	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ ¬ 𝜑) → ¬ 𝜑)
7	5, 6	orim12i 749	. . . . 5 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
8	4, 7	sylbi 120	. . . 4 ⊢ (𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))
9	2, 8	vtoclg 2786	. . 3 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑)))
10	9	pm2.43i 49	. 2 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))
11		df-dc 825	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
12	10, 11	sylibr 133	1 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  DECID wdc 824   = wceq 1343   ∈ wcel 2136  ifcif 3520

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-if 3521

This theorem is referenced by: (None)