Theorem ifmdc 3613

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 2269	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴 ∈ if(𝜑, 𝐵, 𝐶)))
2	1	imbi1d 231	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑)) ↔ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))))
3		df-if 3573	. . . . . 6 ⊢ if(𝜑, 𝐵, 𝐶) = {𝑥 ∣ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑))}
4	3	abeq2i 2317	. . . . 5 ⊢ (𝑥 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑)))
5		simpr 110	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝜑)
6		simpr 110	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ ¬ 𝜑) → ¬ 𝜑)
7	5, 6	orim12i 761	. . . . 5 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
8	4, 7	sylbi 121	. . . 4 ⊢ (𝑥 ∈ if(𝜑, 𝐵, 𝐶) → (𝜑 ∨ ¬ 𝜑))
9	2, 8	vtoclg 2834	. . 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 837	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
12	10, 11	sylibr 134	1 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710  DECID wdc 836   = wceq 1373   ∈ wcel 2177  ifcif 3572

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-if 3573

This theorem is referenced by: (None)